Question

Check whether Lagrange's mean value theorem is applicable on f(x) = sin x + cos x interval $[0,\frac{\pi }{2}]$

Answer 1

$f\left(x\right)=\sin x+\cos x$ in $[0,\frac{\pi }{2}]$

As we know that all trigonometric functions are continuous and differentiable in their domain.

Thus $f\left(x\right)$ is also continuous and differentiable

Thus, the condition of mean value theorem are satisfied.

Hence, there exists atleast one $c\in (0,\pi )$ such that ,

$f^{\prime }\left(c\right)=\frac{[f\left(\frac{\pi }{2}\right)-f\left(0\right)]}{\frac{\pi }{2}-0}$

$\cos c-\sin c=\frac{(\sin \frac{\pi }{2}+\cos \frac{\pi }{2})-(\sin 0+\cos 0)}{(\frac{\pi }{2}-0)}$

$\cos c-\sin c=0$

$\cos c=\sin c$

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

$\frac{\pi }{4}$ lies in the given interval.

Hence LMV is verified.