[Math] Rolle’s theorem and mean value theorem 2 questions

calculusderivatives

How would I solve the following two questions.

Question 1

Determine whether the functions $f(x)=3\sin(2x)$ satisfies conditions of Rolle's theorem for the interval $[0,\pi]$. If so find all numbers $c$ that satisfy the conclusion of the theorem.

This be what I did.

I know $f(x)=3\sin(2x)$ is continuous on $[0,\pi]$ and differentiable on the interval $(0,\pi)$.

I know $f'(c)=0$ in Rolle's theorem.

So I did the derivative and got $3\cos(2c)(2)$=0

and then got $6\cos2c=0$ but how would I get $c$?

Question 2

Determine whether the function $f(x)=1+\sqrt[3]{x^2}$ satisfies conditions of the mean value theorem for the interval $[-1,1]$.

I know $f'(x)=\frac{2}{3}x^{-1/3}$ and for $c$ I got $1$ using $\frac{f(b)-f(a)}{b-a}$

then I set $f'(x)=\frac{2}{3}x^{-1/3}=1$ but how would I get $c$?

Best Answer

Hint:

  1. For $f(x)=3\sin(2x)$ is sufficient to solve equation $\cos{2c}=0 \Rightarrow 2c=\dfrac{\pi}{2} \Rightarrow c=\dfrac{\pi}{4}\;\;\;(c\in(0,\;\pi))$
  2. Derivative of $f(x)=1+\sqrt[3]{x^2}$ is discontinuous at the point $x=0,$ so $f(x)=1+\sqrt[3]{x^2}$ does not satisfy theorem conditions.
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