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 will exist where the second derivative is zero.

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This will never be , so there are no points of inflection.

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.

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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 points of inflection will only exist where the second derivative is zero.

Now  therefore, there are no points of inflection.

The points of inflection, if any exist, will be found where the second derivative is zero.

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To find the y-coordinates, we simply plug the x-coordinates in to the original function.

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So  is an inflection point.

Also, , so  is another inflection point.

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 .

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 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.

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 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,

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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).