Mean Value Theorems — Maths STD 12 Science — Question

Since, $\sin\text{x}$ and $\cos\text{x}$ are everywhere differentiable and continuous,

$\text{f}(\text{x})=\sin\text{x}+\cos\text{x}$ is continuous on $\Big[0,\frac{\pi}{2}\Big]$ and differentiable on $\Big(0,\frac{\pi}{2}\Big)$

Also,

$\text{f}\Big(\frac{\pi}{2}\Big)=\text{f}(0)=1$

Thus, f(x) satisfies all the conditions of Rolle's theorem.

Now, we have show that there must exists $\text{c}\in\Big(0,\frac{\pi}{2}\Big)$ such that f'(c) = 0.

We have

$\text{f}(\text{x})=\sin\text{x}+\cos\text{x}$

$\Rightarrow\text{f}'(\text{x})=\cos\text{x}-\sin\text{x}$

$\therefore\ \text{f}'(\text{x})=0$

$\Rightarrow\cos\text{x}-\sin\text{x}=0$

$\Rightarrow\tan\text{x}=1$

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

Thus, $\text{c}=0\in\Big(0,\frac{\pi}{2}\Big)$ such that f'(c) = 0.

Hence, Rolle's theorem is verified.