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

Jul 22, 2018

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

#### Explanation:

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

$- 1 = 2 \cos \left(x\right)$, and then $\cos \left(x\right) = - \frac{1}{2}$. Between $0$ and $2 \pi$ these points are:

$x = \frac{4}{6} \pi = \frac{2}{3} \pi$ and $\frac{8}{6} \pi = \frac{4}{3} \pi$,

and all the congruents are:

$x = \frac{2}{3} \pi + K \cdot 2 \pi$ and $\frac{4}{3} \pi + K \cdot 2 \pi$ where $K$ is any integer number (positive or negative)

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

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

$x = \frac{2}{3} \pi + K \cdot 2 \pi$ are all local maxima, and the points
$\frac{4}{3} \pi + K \cdot 2 \pi$ are all local minima