Question

Hel[p plssssss ayuda

Answer 1

f is increasing from (-18,0),(0,2)

Explanation:

to figure out when a function is increasing, You take the derivative and figure out when it is positive. For this problem, we are given the derivative function.

`-x^4-16x^3+36x`

TO figure out when this is positive we will find the roots of the equation. TO do this we set it equal to zero.

`-x^4-16x+36x^2=0` The first thing I am going to do is simplify the equation by factoring out a -x^2

`(-x^2)(x^2+16x+36)=0` Next, we can simplify the trinomial thus giving us

`(-x^2)(x+18)(x-2)=0`

Now this can be zero if any of the individual parts are zero so we set each equal to zero and solve for x

`-x^2=0, x+18=0, x-2=0`

This will give us the roots of 0 twice, -18, and 2.

Now that we have the roots we can make a number line filling in the roots on the line

_____-18_____________0________2_____

We know at those specific points the derivative function is zero. What we need to find out now is whether it is positive or negative between those points. Positive indicates an increase and negative indicates a decrease.

So we plug in the number between those roots

ex f'(1) = 19 so positive

f'(3)= -189 so negative

so we fill in the number line

_____-18_________0 +++ 2 ------

Now to finish we pick number representing the other two points

F'(-20)= -17600 so negative

f'(-1)= 51 so positive

this will give a final number line

---- -18 +++++ 0 ++++ 2 ---

this shows that the derivative function is positive from -18 to 0 and from 0 to 2

So f is increasing from (-18,0),(0,2)

Hope this helps