Answer the following questions about the function whose derivative is given below

Question

anonymous · Answer

$$f \prime(x)=(x-5)^{2}(x+7)$$

anonymous · Answer

a. what are the critical points of f?
b. On what intervals is f increasing or decreasing?
c. at what points, if any, does f assume local maximum and minimum values?

agent0smith · Answer

a) To find critical points, let f'(x) = 0 and solve for x
$$\large 0 =(x-5)^{2}(x+7)$$

anonymous · Answer

x=-7, 5

agent0smith · Answer

Correct. Now, to find where f is increasing, that means f' must be positive (ie f' > 0). So check the sign of f' to the left and right of those two points... 

ie find if f'(-7.1) and f'(-6.9) are positive or negative. And find if f'(4.9) and f'(5.1) are positive or negative. Drawing a picture can help.

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 f' = 0 at -7 and 5, so find if the slope is pos/neg to the left and right of those.

anonymous · Answer

where did you get the points?

anonymous · Answer

f'(x) = 0 where there are critical points. solve for x, then plug x back into the original equation to find the y values of all the points

anonymous · Answer

that's where @agent0smith got his points

anonymous · Answer

I don't understand how

agent0smith · Answer

@clavoie ...you already posted the points... 

clavoie Best Response   0
x=-7, 5

which means
f'(-7) = 0 and f'(5) = 0 ie the slope at x=-7 and 5 is zero.

Hence why I said find if f'(-7.1) and f'(-6.9) are positive or negative. 

And find if f'(4.9) and f'(5.1) are positive or negative.

anonymous · Answer

where are you getting the y values?

agent0smith · Answer

What y values... you solved this $$\large 0 =(x-5)^{2}(x+7)$$ for x and got x=-7 and 5... that means f' must be zero at those points, since that equation is f'.

anonymous · Answer

why is it (-7, 1)? where'd the 1 come from?

agent0smith · Answer

it's not (-7, 1). I picked a value to the left of x = -7... i picked -7.1

Reread what I wrote above:

Now, to find where f is increasing, that means f' must be positive (ie f' > 0). So check the sign of f' to the left and right of those two points... 

ie find if f'(-7.1) and f'(-6.9) are positive or negative. And find if f'(4.9) and f'(5.1) are positive or negative. Drawing a picture can help.

anonymous · Answer

okay