Second Derivatives - AP Calculus BC
Card 1 of 400
Over which of the following intervals is the following function concave upward?
Over which of the following intervals is the following function concave upward?
Tap to reveal answer
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when 
. However, keep in mind that the domain of f(x) will include only positive values of x, because lnx is only defined for x > 0.
The answer is { 
}.
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when
The answer is {
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Over which of the following intervals is the following function concave upward?
Over which of the following intervals is the following function concave upward?
Tap to reveal answer
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when . However, keep in mind that the domain of f(x) will include only positive values of x, because lnx is only defined for x > 0.
The answer is { }.
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when
The answer is {
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Over which of the following intervals is the following function concave upward?
Over which of the following intervals is the following function concave upward?
Tap to reveal answer
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when . However, keep in mind that the domain of f(x) will include only positive values of x, because lnx is only defined for x > 0.
The answer is { }.
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when
The answer is {
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Over which of the following intervals is the following function concave upward?
Over which of the following intervals is the following function concave upward?
Tap to reveal answer
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when . However, keep in mind that the domain of f(x) will include only positive values of x, because lnx is only defined for x > 0.
The answer is { }.
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when
The answer is {
← Didn't Know|Knew It →
Over which of the following intervals is the following function concave upward?
Over which of the following intervals is the following function concave upward?
Tap to reveal answer
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when . However, keep in mind that the domain of f(x) will include only positive values of x, because lnx is only defined for x > 0.
The answer is { }.
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when
The answer is {
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Over which of the following intervals is the following function concave upward?
Over which of the following intervals is the following function concave upward?
Tap to reveal answer
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when . However, keep in mind that the domain of f(x) will include only positive values of x, because lnx is only defined for x > 0.
The answer is { }.
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when
The answer is {
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Over which of the following intervals is the following function concave upward?
Over which of the following intervals is the following function concave upward?
Tap to reveal answer
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when . However, keep in mind that the domain of f(x) will include only positive values of x, because lnx is only defined for x > 0.
The answer is { }.
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when
The answer is {
← Didn't Know|Knew It →
Over which of the following intervals is the following function concave upward?
Over which of the following intervals is the following function concave upward?
Tap to reveal answer
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when . However, keep in mind that the domain of f(x) will include only positive values of x, because lnx is only defined for x > 0.
The answer is { }.
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when
The answer is {
← Didn't Know|Knew It →
Over which of the following intervals is the following function concave upward?
Over which of the following intervals is the following function concave upward?
Tap to reveal answer
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when . However, keep in mind that the domain of f(x) will include only positive values of x, because lnx is only defined for x > 0.
The answer is { }.
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when
The answer is {
← Didn't Know|Knew It →
Over which of the following intervals is the following function concave upward?
Over which of the following intervals is the following function concave upward?
Tap to reveal answer
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when . However, keep in mind that the domain of f(x) will include only positive values of x, because lnx is only defined for x > 0.
The answer is { }.
In order to determine the concavity of a function, we need to examine its second derivative.
First, we will take the derivative of the function by applying the product rule.
We will then take the derivative of the first derivative. Again, we will employ the product rule.
A funciton is concave upward whenever its second derivative is positive. Thus, we must find the interval over which f''(x) is positive.
This will only be true when
The answer is {
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What type of point is 
in 
?
What type of point is
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Even though the first derivative (
) is 
at 
, there is no max or min because the function is increasing on both sides (derivative is positive on both sides). However, the function is changing its concavity (from negative to positive) at 
. We can tell because the second derivative (
) is also 
at 
, and it's going from negative to positive. Hole and asymptote are irrelevant. When the second derivative changes signs around a specific point we call this an inflection point. An inflection point describes a point that changes the concavity of a function.
Even though the first derivative (
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Determine the intervals on which the function is concave up:
on the interval 
Determine the intervals on which the function is concave up:
on the interval
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To determine the intervals on which the function is concave up, we must find the intervals on which the second derivative of the function is positive.
First, we must find the second derivative of the function:
The derivatives were found using the following rules:
, 
, 
, 
Next, we find the values on the given interval for which the second derivative is equal to zero:
We now use these values as bounds for the intervals on which we check the sign of the second derivative:
Note that at the bounds of the intervals, the second derivative is neither positive nor negative.
Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is negative, while on the second interval, the second derivative is positive. Thus, the function is concave up on the second interval, 
.
To determine the intervals on which the function is concave up, we must find the intervals on which the second derivative of the function is positive.
First, we must find the second derivative of the function:
The derivatives were found using the following rules:
Next, we find the values on the given interval for which the second derivative is equal to zero:
We now use these values as bounds for the intervals on which we check the sign of the second derivative:
Note that at the bounds of the intervals, the second derivative is neither positive nor negative.
Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is negative, while on the second interval, the second derivative is positive. Thus, the function is concave up on the second interval,
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Tell whether g(w) is concave up or concave down when 
. How do you know?
Tell whether g(w) is concave up or concave down when
Tap to reveal answer
Tell whether g(w) is concave up or concave down when 
. How do you know?
To test for concavity, we need to know the sign of our function's second derivative at the given value.
Let's begin by finding the first derivative. To do so, we need to recall the following:
Those two rules tell us how to derive functions made up of polynomials and exponential terms.
Apply the rules to get the following.
Now, do derive again to get the 2nd derivative.
Now, we are almost there, but we need to find g"(-3)
So, our second derivative is positive when w=-3. This means our original function is concave up at this point.
Concave up, because 
Tell whether g(w) is concave up or concave down when
To test for concavity, we need to know the sign of our function's second derivative at the given value.
Let's begin by finding the first derivative. To do so, we need to recall the following:
Those two rules tell us how to derive functions made up of polynomials and exponential terms.
Apply the rules to get the following.
Now, do derive again to get the 2nd derivative.
Now, we are almost there, but we need to find g"(-3)
So, our second derivative is positive when w=-3. This means our original function is concave up at this point.
Concave up, because
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At which of the following values of 
is the function
concave down?
At which of the following values of
concave down?
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The function is concave down wherever 
, so we compute 
and see where it is negative. We have:
(a parabola, opening upwards)
To find where 
is negative, we first find its zeros by setting 
:
,
so 
when 
or 
, and we conclude that 
is negative (
is concave down) between them. That is, 
. The only answer choice completely inside this interval (not outside or at the endpoints) is
.
The function is concave down wherever
To find where
so
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Find the inflection point(s) of 
.
Find the inflection point(s) of
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Inflection points can only occur when the second derivative is zero or undefined. Here we have
.
Therefore possible inflection points occur at 
and 
. However, to have an inflection point we must check that the sign of the second derivative is different on each side of the point. Here we have
.
Hence, both are inflection points
Inflection points can only occur when the second derivative is zero or undefined. Here we have
Therefore possible inflection points occur at
Hence, both are inflection points
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Below is the graph of 
. How many inflection points does 
have?
Below is the graph of
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Possible inflection points occur when 
. This occurs at three values, 
. However, to be an inflection point the sign of 
must be different on either side of the critical value. Hence, only 
are critical points.
Possible inflection points occur when
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Find the point(s) of inflection for the function 
.
Find the point(s) of inflection for the function
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A point of inflection is found where the graph (or image) of a function changes concavity. To find this algebraically, we want to find where the second derivative of the function changes sign, from negative to positive, or vice-versa. So, we find the second derivative of the given function 
The first derivative using the power rule
is,
and the seconds derivative is 
We then find where this second derivative equals 
. 
when 
.
We then look to see if the second derivative changes signs at this point. Both graphically and algebraically, we can see that the function 
does indeed change sign at, and only at, 
, so this is our inflection point.
A point of inflection is found where the graph (or image) of a function changes concavity. To find this algebraically, we want to find where the second derivative of the function changes sign, from negative to positive, or vice-versa. So, we find the second derivative of the given function
The first derivative using the power rule
We then find where this second derivative equals
We then look to see if the second derivative changes signs at this point. Both graphically and algebraically, we can see that the function
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What are the 
coordinates of the points of inflection for the graph
What are the
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Infelction points are the points of a graph where the concavity of the graph changes. The inflection points of a graph are found by taking the double derivative of the graph equation, setting it equal to zero, then solving for 
.
To take the derivative of this equation, we must use the power rule,
.
We also must remember that the derivative of an constant is 0.
After taking the first derivative of the graph equation using the power rule, the equation becomes
.
In this problem the double derivative of the graph equation comes out to 
, factoring this equation out it becomes 
.
Solving for 
when the equation is set equal to zero, the inflection points are located at 
.
Infelction points are the points of a graph where the concavity of the graph changes. The inflection points of a graph are found by taking the double derivative of the graph equation, setting it equal to zero, then solving for
To take the derivative of this equation, we must use the power rule,
We also must remember that the derivative of an constant is 0.
After taking the first derivative of the graph equation using the power rule, the equation becomes
In this problem the double derivative of the graph equation comes out to
Solving for 
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Find all the points of inflection of
Find all the points of inflection of
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In order to find the points of inflection, we need to find 
using the power rule 
.
Now to find the points of inflection, we need to set 
.
.
Now we can use the quadratic equation.
Recall that the quadratic equation is
,
where a,b,c refer to the coefficients of the equation 
.
In this case, a=12, b=0, c=-4.
Thus the possible points of infection are
.
Now to check if or which are inflection points we need to plug in a value higher and lower than each point. If there is a sign change then the point is an inflection point.
To check 
lets plug in 
.
Therefore 
is an inflection point.
Now lets check 
with 
.
Therefore 
is also an inflection point.
In order to find the points of inflection, we need to find
Now to find the points of inflection, we need to set
Now we can use the quadratic equation.
Recall that the quadratic equation is
where a,b,c refer to the coefficients of the equation
In this case, a=12, b=0, c=-4.
Thus the possible points of infection are
Now to check if or which are inflection points we need to plug in a value higher and lower than each point. If there is a sign change then the point is an inflection point.
To check
Therefore
Now lets check
Therefore
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Find all the points of infection of
.
Find all the points of infection of
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In order to find the points of inflection, we need to find 
using the power rule .
Now lets factor 
.
Now to find the points of inflection, we need to set .
.
From this equation, we already know one of the point of inflection, 
.
To figure out the rest of the points of inflection we can use the quadratic equation.
Recall that the quadratic equation is
, where a,b,c refer to the coefficients of the equation
.
In this case, a=20, b=0, c=-18.
Thus the other 2 points of infection are
To verify that they are all inflection points we need to plug in values higher and lower than each value and see if the sign changes.
Lets plug in 
Since there is a sign change at each point, all are points of inflection.
In order to find the points of inflection, we need to find
Now lets factor
Now to find the points of inflection, we need to set
From this equation, we already know one of the point of inflection,
To figure out the rest of the points of inflection we can use the quadratic equation.
Recall that the quadratic equation is
In this case, a=20, b=0, c=-18.
Thus the other 2 points of infection are
To verify that they are all inflection points we need to plug in values higher and lower than each value and see if the sign changes.
Lets plug in
Since there is a sign change at each point, all are points of inflection.
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