The critical numbers of a function are the points at which its slope is zero, which tells where the function has a minimum or maximum. The slope of a function is described by its derivative, so we'll take the derivative of the function and set it equal to 0 to solve for the x values of the critical numbers:

It appears the derivative cannot be factored to solve for x, so we'll have to use the quadratic formula to find the critical numbers:

So the critical numbers occur at the following two values of x:

Critical numbers are where the slope of the function is equal to zero or undefined.

Find the derivative and set the derivative function to zero.

There is only one critical value at .

The critical numbers are the  values for which either

    or     is undefined.

In order to find the first derivative we use the power rule which states

Applying this rule term by term we get

The first derivative is defined for all values of x. Setting the first derivative to zero yields

As such, the critical number is

To solve, simply find the first derivative and find when it is equal to 0. To find the first derivative, we must use the power rule as outlind below.

Power rule:

Thus,

Now, we must set out function equal to 0 and solve for x. Thus,

Dividing both sides by 2, we get:

To solve, simply differentiate using the power rule, as outlined below.

Power rule states,

 .

Thus given,

our first derivative is:

Then plug in 0 for f(x) to find when our function is equal to 0.

Thus,

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 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:

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 point of inflection exists where the second derivative is zero.

, and we set this equal to zero.