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Let f(x)= x^3+3x^2-24x+23

 

** I ALREADY GOT THE FIRST PART. IT SAYS USE DEFINITION OF A DERIVATIVE to find f'(x)= 3x^2+6x-24

 

please help with these::::

b) use the definition of a derivative to find f '' (x)= _____??

c) on what interval is f increasing (include the endpoints in the interval)=___???

d) interval f decreasing =_____???

e) what interval is f concave downward???____?

f) interval is f convave upard???____

 Jun 11, 2020
 #1
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You can use the definition of derivative on \(f'(x)\) to find \(f''(x)\) because \(f''(x) = \displaystyle\lim_{h\to0}\dfrac{f'(x + h) - f'(x)}{h}\).

 

If f is increasing on an interval \(\mathcal I\), then for all real numbers \(x\in \mathcal I\)\(f'(x) \geqslant 0\).

 

If f is decreasing on an interval \(\mathcal I\), then for all real numbers \(x\in \mathcal I\)\(f'(x) \leqslant 0\).

 

If f is concave downward on an interval \(\mathcal I\), then for all real numbers \(x\in \mathcal I\)\(f''(x) \leqslant 0\).

 

If f is concave upward on an interval \(\mathcal I\), then for all real numbers \(x\in \mathcal I\)\(f''(x) \geqslant 0\).

 

These are the definitions. I believe you can finish the rest with these.

 Jun 11, 2020

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