Question

verify mean value theorem for function 2sinx+sin2x on [0,pi]

Answer 1

The Mean Value Theorem says that
if a function f is
continuous on the closed interval [a,b] and
differentiable on the open interval (a,b),

then there is a number c in (a,b) with

such that f'(c)=(f(b)−f(a))/
(b−a.)

In the question:
f(x)=2sinx+sin2x the interval is [0,π]and (0,π) .

It is true that:
f is continuous at every real number, so it is continuous on [0,π]
(sin is continuous, and 2x is continuous so sin2xis contiunuous. And the sum of continuous functions is continuous.)

It is also true that
f is differentiable at every real number, so it is differentiable on (0,π).
(Differentiable means the derivative exists and f'(x)=2cosx+2cos2x exists (is defined) for all values of x.)

Theerfore, the Mean Value Theorem allows us to conclude that:

There is a number c in (0,π) with:

such that
f'(c)=(f(0)−f(π))/(0-π)

f(0)=2sin(0)+ sin(0)
=0
f(π)=2sinπ+sin2π
=0

We can conclude that
There is a number c in (4π,5π) with:

such that 2cosc+2cos2c= 0

That conclude the use of the Mean Value Theorem for f(x)=2sinx+sin2x on the interval is [0,π].
Read it carefully and try to understand it.