Question

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

Answer 1

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'(c) = 0` or `f'(c)` does not exist.

This question:

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

Domain of `f` is `(-oo,oo)`

`f'(x) = 6+3cos(3x)`

`f'(x)` exists for all `x`.

`x` is a solution of `6+3cos(3x) = 0` if and only if

`cos(3x) = -2`.

But this has no real solutions.

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