Verify Rolle's theorem of the following function on the indicated…

Question

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=4^{\sin\text{x}}\text{ on }[0,\pi]$

✓

Answer

Here, $\text{f}(\text{x})=4^{\sin\text{x}}\text{ on }[0,\pi]$We know that exponential and $\sin\text{x}$ both are continuous and differentiable, so f(x) is continuous is $[0,\pi]$ and differentiable is $(0,\pi).$
Now,
$\text{f}(0)=4^{\sin0}=4^0=1$ $\text{f}(\pi)=4^{\sin\pi}=4^0=1$ $\Rightarrow\text{f}(0)=\text{f}(\pi)$ So, Rolle's theorem is applicable, there must exist a point $\text{c}\in(0,\pi)$ such that f'(c) = 0. Now, $\text{f}(\text{x})=4^{\sin\text{x}}$ $\text{f}'(\text{x})=4^{\sin\text{x}}\log4\times\cos\text{x}$ Now, $\text{f}'(\text{c})=0$ $4^{\sin\text{c}}\times\cos\times\text{c}\log4=0$ $\Rightarrow\cos\text{c}=0$ $\Rightarrow\text{c}=\frac{\pi}{2}\in(0,\pi)$ Hence, Rolle's theorem is verified.

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