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≈ and it occurs at x≈ - 0.41
(round to two decimal places.)

Explanation:

Step1: Find derivative of $f(x)$

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

Step2: Set $f'(x) = 0$

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

Step3: Evaluate $f(x)$ at critical points

$f(-0.41)=2(-0.41)^{3}-(-0.41)\approx0.28$
$f(0.41)=2(0.41)^{3}-0.41\approx - 0.28$

Answer:

The local maximum is $y\approx0.28$ and it occurs at $x\approx - 0.41$.
The local minimum is $y\approx - 0.28$ and it occurs at $x\approx0.41$.
$f(x)$ is increasing on $(-\infty,-\frac{1}{\sqrt{6}})\cup(\frac{1}{\sqrt{6}},\infty)$ and decreasing on $(-\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}})$.