Determine Points of Inflection - Pre-Calculus
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Find the x-coordinates of all points of inflection of the function 
.
Find the x-coordinates of all points of inflection of the function
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We set the second derivative of the function equal to zero to find the x-coordinates of any points of inflection.
, and the quadratic formula yields
.
We set the second derivative of the function equal to zero to find the x-coordinates of any points of inflection.
Determine the location of the points of inflection for the following function:
Determine the location of the points of inflection for the following function:
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The points of inflection of a function are the points at which its concavity changes. The concavity of a function is described by its second derivative, which will be equal to zero at the inflection points, so we'll start by finding the first derivative of the function:
Next we'll take the derivative one more time to get the second derivative of the original function:
Now that we have the second derivative of the function, we can set it equal to 0 and solve for the points of inflection:
The points of inflection of a function are the points at which its concavity changes. The concavity of a function is described by its second derivative, which will be equal to zero at the inflection points, so we'll start by finding the first derivative of the function:
Next we'll take the derivative one more time to get the second derivative of the original function:
Now that we have the second derivative of the function, we can set it equal to 0 and solve for the points of inflection:
Which of the following is an 
-coordinate of an inflection point of the graph of the following function?
Which of the following is an
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The inflection points of a function are the points where the concavity changes, either from opening upwards to opening downwards or vice versa. The inflection points occur at the x-values where the second derivative is either zero or undefined. That means we need to find our second derivative.
We start by using the Power Rule to find the first derivative.
Taking the derivative once more gives the second derivative.
We then set this derivative equal to zero and solve.
This factors nicely.
Therefore our second derivative is zero when 
8 is the only one of these two amongst our choices and is therefore our answer.
The inflection points of a function are the points where the concavity changes, either from opening upwards to opening downwards or vice versa. The inflection points occur at the x-values where the second derivative is either zero or undefined. That means we need to find our second derivative.
We start by using the Power Rule to find the first derivative.
Taking the derivative once more gives the second derivative.
We then set this derivative equal to zero and solve.
This factors nicely.
Therefore our second derivative is zero when
8 is the only one of these two amongst our choices and is therefore our answer.
Determine the points of inflection of the following function:
Determine the points of inflection of the following function:
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The points of inflection of a function are those at which its second derivative is equal to 0. First we find the second derivative of the function, then we set it equal to 0 and solve for the inflection points:
The points of inflection of a function are those at which its second derivative is equal to 0. First we find the second derivative of the function, then we set it equal to 0 and solve for the inflection points:
Find the inflection points of the following function:
Find the inflection points of the following function:
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The points of inflection of a function are those at which its second derivative is equal to 0. First we find the second derivative of the function, then we set it equal to 0 and solve for the inflection points:
The points of inflection of a function are those at which its second derivative is equal to 0. First we find the second derivative of the function, then we set it equal to 0 and solve for the inflection points:
Find the points of inflection of the following function:
Find the points of inflection of the following function:
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The points of inflection of a function are those at which its second derivative is equal to 0. First we find the second derivative of the function, then we set it equal to 0 and solve for the inflection points:
The points of inflection of a function are those at which its second derivative is equal to 0. First we find the second derivative of the function, then we set it equal to 0 and solve for the inflection points:
Determine the points of inflection, if any, of the following function:
Determine the points of inflection, if any, of the following function:
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The points of inflection of a function are those at which its concavity changes. The concavity of a function is described by its second derivative, and when the second derivative is 0 a point of inflection occurs. We find the second derivative of the function and then set it equal to 0 to solve for the inflection points:
So the function has only one point of inflection at x=5/3.
The points of inflection of a function are those at which its concavity changes. The concavity of a function is described by its second derivative, and when the second derivative is 0 a point of inflection occurs. We find the second derivative of the function and then set it equal to 0 to solve for the inflection points:
So the function has only one point of inflection at x=5/3.
Determine the x-coordinate of the inflection point of the function 
.
Determine the x-coordinate of the inflection point of the function
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The point of inflection exists where the second derivative is zero.
, and we set this equal to zero.
The point of inflection exists where the second derivative is zero.
Find the point of inflection of the function 
.
Find the point of inflection of the function
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To find the x-coordinate of the point of inflection, we set the second derivative of the function equal to zero.
.
To find the y-coordinate of the point, we plug the x-coordinate back into the original function.
The point is then 
.
To find the x-coordinate of the point of inflection, we set the second derivative of the function equal to zero.
To find the y-coordinate of the point, we plug the x-coordinate back into the original function.
The point is then
List the intervals and determine where the graph of 
is concave up and concave down.
List the intervals and determine where the graph of
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We need to set the second derivative equal to zero to determine where the inflection points are.
are the x-coordinates of our inflection points. Thus the intervals of concavity are 
, 
, and 
. We can use 
as our test points.
, so the graph is concave up on 
.
, so the graph is concave down on 
.
, so the graph is concave up on 
.
We need to set the second derivative equal to zero to determine where the inflection points are.
Determine the point(s) of inflection of 
.
Determine the point(s) of inflection of
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The points of inflection exist where the second derivative is zero.
which can never be 
. Therefore, there are no points of inflection.
The points of inflection exist where the second derivative is zero.
Determine the x-coordinate(s) of the point(s) of inflection of the function 
.
Determine the x-coordinate(s) of the point(s) of inflection of the function
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Any points of inflection that exist will be found where the second derivative is equal to zero.
.
Since 
, we can focus on 
. Thus
, and 
.
Any points of inflection that exist will be found where the second derivative is equal to zero.
Since
Find the point(s) of inflection of 
.
Find the point(s) of inflection of
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The points of inflection will exist where the second derivative is zero.
.
This will never be 
, so there are no points of inflection.
The points of inflection will exist where the second derivative is zero.
This will never be
List the interval(s) where the function 
is concave up.
List the interval(s) where the function
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The graph is concave up where the second derivative is positive. Let us first find out if there are any points of inflection to narrow our search.
.
Now we can perform the second derivative concavity test on points on either side of 
. Let us try 
.
The second derivative at 
gives us 
which is less than zero, so the graph is concave down in this interval. The second derivative at 
gives us 
, which is positive. Hence, the graph is concave up on the interval 
.
The graph is concave up where the second derivative is positive. Let us first find out if there are any points of inflection to narrow our search.
Now we can perform the second derivative concavity test on points on either side of
The second derivative at
Find the x-coordinate(s) of the point(s) of inflection of 
.
Find the x-coordinate(s) of the point(s) of inflection of
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The points of inflection will only exist where the second derivative is zero.
Now 
therefore, there are no points of inflection.
The points of inflection will only exist where the second derivative is zero.
Now
Find the inflection point(s) of 
.
Find the inflection point(s) of
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The points of inflection, if any exist, will be found where the second derivative is zero.
.
To find the y-coordinates, we simply plug the x-coordinates in to the original function.
.
So 
is an inflection point.
Also, 
, so 
is another inflection point.
The points of inflection, if any exist, will be found where the second derivative is zero.
To find the y-coordinates, we simply plug the x-coordinates in to the original function.
So
Also,
Find the x-coordinate(s) of the point(s) of inflection of 
.
Find the x-coordinate(s) of the point(s) of inflection of
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The inflection points, if they exist, will occur where the second derivative is zero.
The inflection points, if they exist, will occur where the second derivative is zero.
Find the point(s) of inflection of the function 
.
Find the point(s) of inflection of the function
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The point of inflection will exist where the second derivative equals zero.
.
Now we need the y-coordinate of the point.
Thus the inflection point is at 
.
The point of inflection will exist where the second derivative equals zero.
Now we need the y-coordinate of the point.
Thus the inflection point is at
Determine the points of inflection for the following function.
Determine the points of inflection for the following function.
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To find inflection points, take the second derivative and set it equal to 
.
Since 
is never 
, this function does not have any inflection points. Thus, the answer is none.
To find inflection points, take the second derivative and set it equal to
Since
Find the point(s) of inflection of the following function:
Find the point(s) of inflection of the following function:
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To solve, simply differentiate twice, find when the function is equal to 0, and then plug into the first equation.
For this particular function, use to power rule to differentiate.
The power rule states,
.
Also recall that the derivative of a constant is zero.
Applying the power rule and rule of a constant once to the function we can find the first derivative.
Thus,
From here apply the power rule and rule of a constant once more to find the second derivative of the function.
Now to solve for the inflection point, set the second derivative equal to zero.
From here plug this x value into the original function to find the y value of the inflection point.
Thus, our point is (0,1).
To solve, simply differentiate twice, find when the function is equal to 0, and then plug into the first equation.
For this particular function, use to power rule to differentiate.
The power rule states,
Also recall that the derivative of a constant is zero.
Applying the power rule and rule of a constant once to the function we can find the first derivative.
Thus,
From here apply the power rule and rule of a constant once more to find the second derivative of the function.
Now to solve for the inflection point, set the second derivative equal to zero.
From here plug this x value into the original function to find the y value of the inflection point.
Thus, our point is (0,1).