Solution assignment 06 Optimizing f(x)

Find the extremes of the function:

$y=\cos(x)$

on the interval $[0,2\pi]$ . Verify whether they are maxima or minima by using the second derivative.

The derivative of the function is:

$y'=-\sin(x)$

The equation:

$-\sin(x)=0$

has the following solutions on the given interval:

$x=0$ , $x=\pi$ or $x=2\pi$

We calculate the second derivative:

$y''=-\cos(x)$

For $x=0$ we have $y''=-1$ and thus there is maximum in this point.

For $x=\pi$ we have $y''=1$ and thus there is minimum in this point.

Voor $x=2\pi$ we have $y''=-1$ and thus there is maximum in this point.