Let's analyze the problem step by step:

Problem Details

The problem involves a function $f$ that is differentiable on $\mathbb{R}$, and we are given:

1. $f(0) = 0$,
2. $f(1) = f(2) = 2$.

We are tasked to prove three results using the Mean Value Theorem (MVT) or similar ideas.

Solutions

Part (i): Show that $\exists c_1 \in (0, 1)$ such that $f'(c_1) = 2$.

Using the Mean Value Theorem, for $f$ on the interval $[0, 1]$:

• $f$ is differentiable on $(0, 1)$ (given),
• $f$ is continuous on $[0, 1]$ (differentiability implies continuity).

By MVT, there exists $c_1 \in (0, 1)$ such that: $f'(c_1) = \frac{f(1) - f(0)}{1 - 0}.$ Substitute the values: $f'(c_1) = \frac{2 - 0}{1 - 0} = 2.$ Thus, such a $c_1$ exists.

Part (ii): Show that $\exists c_2 \in (1, 2)$ such that $f'(c_2) = 0$.

Similarly, apply the MVT to $f$ on the interval $[1, 2]$:

• $f$ is differentiable on $(1, 2)$,
• $f$ is continuous on $[1, 2]$.

By MVT, there exists $c_2 \in (1, 2)$ such that: $f'(c_2) = \frac{f(2) - f(1)}{2 - 1}.$ Substitute the values: $f'(c_2) = \frac{2 - 2}{2 - 1} = 0.$ Thus, such a $c_2$ exists.

Part (iii): Show that $\exists c_3 \in (0, 2)$ such that $f'(c_3) = 1$.

Here, we use the Intermediate Value Theorem for Derivatives (a consequence of the Mean Value Theorem).

• From part (i), $f'(c_1) = 2$ for some $c_1 \in (0, 1)$,
• From part (ii), $f'(c_2) = 0$ for some $c_2 \in (1, 2)$.

Since $f'$ is continuous (because $f$ is differentiable), by the Intermediate Value Theorem applied to $f'$ on $[c_1, c_2]$, every value between $f'(c_1) = 2$ and $f'(c_2) = 0$ must be attained. In particular, $f'(c_3) = 1$ for some $c_3 \in (0, 2)$.

Summary

The required points $c_1$, $c_2$, and $c_3$ exist by the application of the Mean Value Theorem and the Intermediate Value Theorem for derivatives.

Do you want further clarifications on any of the parts, or additional examples? Here are some related questions:

1. What is the Mean Value Theorem, and why is it useful in calculus?
2. How does the Intermediate Value Theorem for derivatives extend the Mean Value Theorem?
3. Can we always apply the Mean Value Theorem to any function?
4. What are the implications of differentiability on a function’s continuity?
5. How can the results be generalized for other intervals or functions?

Tip: Always verify that a function satisfies the conditions (differentiability and continuity) before applying the Mean Value Theorem.