Verify Lagranges mean value theorem for the following functions on the indicated intervals. `f(x)=x-2sinxon[-pi,pi]` `f(x)=2sinx+sin2xon[0,pi]` `f(x)=(log)_e xon[1,2]` `f(x)={2+x^3,ifxlt=1 3x ,x >0on[-1,2]`

Answer

Proved

Explanation

ATQ,

`f}(x})=2 \sin x}+\sin 2 x}, x} \in[0, \pi]`


`f(x)` is continuous is `[0, \pi]`

`f}(x})` is differentiable in `(0, \pi)`


Thus, both the conditions of Lagrange's man value theorem are satisfied by the function `f(x)` in `[0, \pi]`, therefore, there exists at least one real number c in `[0, \pi]` such that


$$
\begin{aligned}
&f^{\prime}(c)=\frac{f(\pi)-f(0)}{\pi-0} \\
...