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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

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

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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