# Critical Numbers/Extrema of a function

• Apr 22nd 2009, 01:53 PM
zaurm
Critical Numbers/Extrema of a function
• Apr 22nd 2009, 02:24 PM
Soroban
Hello, zaurm!

The problem is straight-forward.

Quote:

a) Find the critical numbers of: . $f(x) \:=\:\frac{1}{2}\,x - \sin x$ .in the interval $[0,\pi]$
Equate the derivative to zero and solve . . .

. . $f'(x) \:=\:\frac{1}{2}-\cos x \:=\:0 \quad\Rightarrow\quad \cos x \:=\:\tfrac{1}{2}\quad\Rightarrow\quad x \:=\:\frac{\pi}{3}$

Quote:

b) Find the exrema of $f(x)$ .in the inverval $[0,\pi]$
. . $\begin{array}{ccccccccc}f(0) &=& 0 - \sin 0 &=& 0 \\
f(\frac{\pi}{3}) &=& \frac{1}{2}\!\cdot\!\frac{\pi}{3} - \sin\frac{\pi}{3} &=& \frac{\pi}{6} - \frac{\sqrt{3}}{2} & \leftarrow \text{ minimum}\\
f(\pi) &=& \pi -\sin\pi &=& \pi & \leftarrow \text{ maximum}
\end{array}$