Question

Find the absolue maximum value and the absolute minimum value of the following function in the given intervals:

$f\left(x\right)=sinx+cosx,xϵ[0,\pi ]$

Answer 1

Given function $f\left(x\right)=sinx+cosx\Rightarrow f^{\prime }\left(x\right)=cosx-sinx$
For maxima or minima put $f^{\prime }\left(x\right)=0\Rightarrow cos-sinx=0\Rightarrow \frac{sinx}{cosx}=1$
$\Rightarrow tanx=1\Rightarrow \frac{\pi }{4}ϵ[0,\pi ]$
Now, we evaluate the value of f at critical point $x=\frac{\pi }{4}$ and at the end points of the interval $[0,\pi ]$.
$\begin{array}{cc}Atx=\frac{\pi }{4}& f\left(\frac{\pi }{4}\right)=sin\frac{\pi }{4}+cos\frac{\pi }{4}=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\sqrt{2}\\ Atx=0,& f\left(0\right)=sin0+cos0=0+1=1\\ Atx=\pi & f\left(\pi \right)=sin\pi +cos\pi =0-1=-1\end{array}$
Thus, absolute maximum value is~ $\sqrt{2}atx=\frac{\pi }{4}$ and absolute minimum value is -1 at $x=\pi$