Find the critical numbers of the function. (enter your answers as a comma-separated list. if an answer does not exist, enter dne.) f(x) = x4/5(x − 2)2

The critical values of the function are x = 0 and x = 4.

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In this problem, the function is:

The derivative is found as follows, applying the quotient rule:

The zeros of the function are the zeros of the numerator, thus:

The critical values are x = 0 and x = 4.

A similar problem is given at link

The critical numbers/values are x = 0, 4/7, 2

Step-by-step explanation:

This is a doozy; no wonder you have it up here for help!

The critical numbers of a function are found where the derivative of the function is equal to 0.  To find these numbers, you have to factor the deriative or simply solve it for 0.  This one is especially difficult since it involves rational exponents that have to be factored.  But this is fun, so let's get to it.

First off, I am assuming that the function is

which involves using the product rule to find the derivative.

That derivative is

which simplifies down to

and

Let's get everything over the common denominator of 5 so we can easily add and subtract like terms:

Combining like terms gives us

This, however, factors so it is easier to solve for x.  First we will set this equal to 0, then we will factor out

:

By the Zero Product Property, one of those terms has to equal 0 for the whole product to equal 0.  So

when x = 0

And

when x = 2 and x = 4/7

Those are the critical numbers/values for that function.  This indicates where there is a max value or a min value.

Step-by-step explanation: