Question

Find the greatest value of function Find the greatest and the least values of the following functions.
sin x+cos x

Answer 1

Given function $f\left(x\right)=sinx+cosx$

$f^1\left(x\right)=cosx-sinx$

$f^1\left(x\right)=0$

$\Rightarrow cosx-sinx=0$

$\Rightarrow tanx=1$

$\Rightarrow x=\frac{\pi }{4}$

Finding values at $x=0,\frac{\pi }{4},\pi$

$f\left(0\right)=\cos 0+\sin 0=1+0=1$

$f\left(\frac{\pi }{4}\right)=\cos \left(\frac{\pi }{4}\right)+\sin \left(\frac{\pi }{4}\right)=\sqrt{2}$

$f\left(\pi \right)=\cos \pi +\sin \pi =-1+0=-1$

Absolute minimum occurs at $x=\pi$ and minimum value $=-1$

Absolute maximum occurs at $x=\frac{\pi }{4}$ and max value $\sqrt{2}$.