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

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Answer 1 · 2017-07-29T01:29:09

There are none.

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.