How do you find the critical numbers for $f(x) = x + 2sinx$ to determine the maximum and minimum?

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Answer 1 · 2018-07-22T16:03:12

The critical points of a function $f(x)$ are the $x$ that make $f'(x)=0$

We calculate the derivative $f'(x)=1+2cos(x)$ , and now we need to find where $f'(x)=1+2cos(x)=0$ . But that means:

$-1=2cos(x)$ , and then $cos(x) = -1/2$ . Between $0$ and $2pi$ these points are:

$x=4/6pi=2/3pi$ and $8/6pi=4/3pi$ ,

and all the congruents are:

$x=2/3pi+K*2pi$ and $4/3pi+K*2pi$ where $K$ is any integer number (positive or negative)

Now the second derivative of $f(x)$ is:

$f''(x) = -2sin(x)$ which is negative in the first set of points and positive in the second set. So the points:

$x=2/3pi+K*2pi$ are all local maxima, and the points
$4/3pi+K*2pi$ are all local minima

How do you find the critical numbers for f(x) = x + 2sinxf(x) = x + 2sinx to determine the maximum and minimum?