Question

Let 𝑓(𝑥) = 1 3 𝑥 3 + 𝑥 2 − 15𝑥 − 9 . Use detailed sign tables in answering the following questions. (a) Find the intervals in which 𝑓 is increasing or decreasing. (b) Find the intervals in which the graph of 𝑦 = 𝑓(𝑥) is concave upwards or downwards. 

Answer 1

$f(x)=13x^3+x^2-15x-9$

a)

$f'(x)=39x^2+2x-15$

$f'(x)=0$ when $x=\frac{-2\pm\sqrt{4+4*15*39}}{2*39}=\frac{-1\pm\sqrt{586}}{39}$

$f(x)$ is increasing when $f'(x)>0$ or when

$-\infty<x<\frac{-1-\sqrt{586}}{39}$

and when

$\frac{-1+\sqrt{586}}{39}<x<+\infty$

$f(x)$ is decreasing when $f'(x)<0$ or when

$\frac{-1-\sqrt{586}}{39}<x<\frac{-1+\sqrt{586}}{39}$

b)

$f''(x)=78x+2$

$f''(x)=0$ when $x=-\frac{1}{39}$

When $x<-\frac{1}{39}$ , $f''(x)<0$ , $f(x)$ is concave upwards.

When $x>-\frac{1}{39}$ , $f''(x)>0$ , $f(x)$ is concave downwards.