Solution assignment 05 Optimizing f(x)

Find the extremes of the function:

$y=\sin(x)$

on the interval interval $[0,2\pi]$ .

The derivative of the function is:

$y'=\cos(x)$

The equation:

$\cos(x)=0$

has the following solutions in the given interval:

$x=\displaystyle\frac{\pi}{2}$ or $x=\displaystyle\frac{3\pi}{2}$

When we look at the graph of:

$y=\cos(x)$

we notice that the sign of $y'$ around $\displaystyle\frac{\pi}{2}$ is: + $0$ - and thus there is a maximum. Similarly, the other point is a minimum.