# How do you find all local maximum and minimum points using the second derivative test given y=6x+sin3x?

Jul 29, 2017

There are none.

#### Explanation:

Method:

Find critical numbers, the test the critical numbers using the second derivative test.

A critical number for function $f$ is a number $c$ in the domain of $f$ at which $f ' \left(c\right) = 0$ or $f ' \left(c\right)$ does not exist.

This question:

#f(x) = 6x+sin(3x)

Domain of $f$ is $\left(- \infty , \infty\right)$

$f ' \left(x\right) = 6 + 3 \cos \left(3 x\right)$

$f ' \left(x\right)$ exists for all $x$.

$x$ is a solution of $6 + 3 \cos \left(3 x\right) = 0$ if and only if

$\cos \left(3 x\right) = - 2$.

But this has no real solutions.

Therefore there are no critical numbers for $f$, so there are no local extreme points.