Question

(a) use a graphing utility to graph f(x)=2x^3 - x on the interval -2,2 and approximate any local maxima and local minima.
(b) determine where f is increasing and where it is decreasing.
(a) using a graphing utility, graph the function for -2≤x≤2 and -7≤y≤7. choose the correct graph, below.
the local maximum is y≈0.19 and it occurs at x≈ - 0.41. (round to two decimal places.)

Explanation:

Step1: Find derivative

$f'(x)=6x^{2}-1$

Step2: Set derivative to 0

$6x^{2}-1 = 0$, so $x=\pm\frac{1}{\sqrt{6}}\approx\pm0.41$

Step3: Test intervals

For $x\in(-2,-0.41)$, $f'(x)>0$, increasing. For $x\in(-0.41,0.41)$, $f'(x)<0$, decreasing. For $x\in(0.41,2)$, $f'(x)>0$, increasing.

Answer:

Local maximum is $y\approx0.19$ at $x\approx - 0.41$. Local minimum is $y\approx - 0.19$ at $x\approx0.41$. $f$ is increasing on $(-2,-0.41)\cup(0.41,2)$ and decreasing on $(-0.41,0.41)$.