Mathematics, 16.07.2019 17:20 Geo777

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

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Answer from: gabbihardy7980

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

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The critical values of a function f(x) are the values of x for which $f^{\prime}(x) = 0$ .

In this problem, the function is:

$f(x) = \frac{x^4}{5(x - 2)^2}$

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

$f^{\prime}(x) = \frac{5[x^4]^{\prime}(x-2)^2 - 5x^4[(x-2)^2]^{\prime}}{[5(x - 2)^2]^2}$

$f^{\prime}(x) = \frac{2x^3(x - 4)}{5(x - 2)^3}$

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

2x^3(x - 4) = 0

$2x^3 = 0 \rightarrow x = 0$

$x - 4 = 0 \rightarrow x = 4$

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

A similar problem is given at link

Answer from: jholland03

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

$f(x)=x^{\frac{4}{5}}*(x-2)^2$ which involves using the product rule to find the derivative.

That derivative is

$f'(x)=x^{\frac{4}{5}}*2(x-2)+\frac{4}{5}x^{-\frac{1}{5}}(x-2)^2$ which simplifies down to

$f'(x)=x^{\frac{4}{5}}(2x^{\frac{5}{5}}-4)+\frac{4}{5}x^{-\frac{1}{5}}(x^{\frac{10}{5}}-4x^{\frac{5}{5}}+4)$ and

$f'(x)=2x^{\frac{9}{5}}-4x^{\frac{4}{5}}+\frac{4}{5}x^{\frac{9}{5}}-\frac{16}{5}x^{\frac{4}{5}}+\frac{16}{5}x^{-\frac{1}{5}}$

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

$f'(x)=\frac{10}{5}x^{\frac{9}{5}}-\frac{20}{5}x^{\frac{4}{5}}+\frac{4}{5}x^{\frac{9}{5}}-\frac{16}{5}x^{\frac{4}{5}}+\frac{16}{5}x^{-\frac{1}{5}}$

Combining like terms gives us

$f'(x)=\frac{14}{5}x^{\frac{9}{5}}-\frac{36}{5}x^{\frac{4}{5}}+\frac{16}{5}x^{-\frac{1}{5}}$

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

$\frac{2}{5}x^{-\frac{1}{5}}$ :

$0=\frac{2}{5}x^{-\frac{1}{5}}(7x^2-18x+8)$

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

$\frac{2}{5}x^{-\frac{1}{5}}=0$ when x = 0

And

7x^2-18x+8=0 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.

Answer from: Quest

d, x=8

step-by-step explanation:

simplifying

4x + -5 = 2x + 11

reorder the terms:

-5 + 4x = 2x + 11

reorder the terms:

-5 + 4x = 11 + 2x

solving

-5 + 4x = 11 + 2x

solving for variable 'x'.

move all terms containing x to the left, all other terms to the right.

add '-2x' to each side of the equation.

-5 + 4x + -2x = 11 + 2x + -2x

combine like terms: 4x + -2x = 2x

-5 + 2x = 11 + 2x + -2x

combine like terms: 2x + -2x = 0

-5 + 2x = 11 + 0

-5 + 2x = 11

add '5' to each side of the equation.

-5 + 5 + 2x = 11 + 5

combine like terms: -5 + 5 = 0

0 + 2x = 11 + 5

2x = 11 + 5

combine like terms: 11 + 5 = 16

2x = 16

divide each side by '2'.

Answer from: Quest

its a trick question

step-by-step explanation:

dogs love water, just turn on the hose and it will run in circles chasing the water all day : )

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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

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