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

Question

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\cos2{\text{x}}\text{ on }[0,\pi]$

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Answer

Here$\text{f}(\text{x})=\cos2{\text{x}}\text{ on }[0,\pi]$
We know that, cosine function is continuous and differentiable every where, so f(x) is continuous is $[0,\pi]$ and differentiable is $(0,\pi).$
Now,
$\text{f}(0)=\cos0=1$
$\text{f}(\pi)=\cos(2\pi)=1$
$\Rightarrow\text{f}(0)=\text{f}(\pi)$
So, Rolle's theorem is applicable, so there must exist a point $\text{c}\in(0,\pi)$ such that f'(c) = 0.
Now,
$\text{f}(\text{x})=\cos2\text{x}$
$\text{f}'(\text{x})=-2\sin2\text{x}$
So, $\text{f}'(\text{c})=0$
$\Rightarrow-2\sin2\text{c}=0$
$\Rightarrow\sin2\text{c}=0$
$\Rightarrow2\text{c}=0$ or $2\text{c}=\pi$
$\Rightarrow\text{c}=0$ or $\text{c}=\frac{\pi}{2}\in(0,\pi)$
Hence, Rolle's theorem is verified.

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