The problem consists of two parts:

Part (a):

To prove that $f'(x)$ is odd, given that $f(x)$ is an even function.

Part (b):

Using $f(x) = \ln(x+1)$, prove using the Mean Value Theorem (MVT) that $\ln(x+1) \leq x$ for all $x \geq 0$ and that equality $\ln(x+1) = x$ holds when $x = 0$.

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Solution:

Part (a):

1. Definition of an Even Function: A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in its domain.

2. Derivative of Both Sides: Differentiate $f(-x) = f(x)$ with respect to $x$: [ \frac{d}{dx}[f(-x)] = \frac{d}{dx}[f(x)]. ]

3. Using the Chain Rule: The left-hand side becomes: $\frac{d}{dx}[f(-x)] = f'(-x) \cdot (-1) = -f'(-x).$ The right-hand side is $f'(x)$.

   Thus, we have: $-f'(-x) = f'(x).$

4. Conclusion: Multiplying through by $-1$, we get: $f'(-x) = -f'(x).$ This is the definition of an odd function. Hence, $f'(x)$ is odd.

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Part (b):

1. Mean Value Theorem (MVT): The Mean Value Theorem states that for a continuous and differentiable function $f(x)$ over $[a, b]$, there exists $c \in (a, b)$ such that: $f'(c) = \frac{f(b) - f(a)}{b - a}.$

2. Apply MVT to $f(x) = \ln(x+1)$: Let $f(x) = \ln(x+1)$. Consider the interval $[0, x]$ for $x > 0$: $f'(c) = \frac{\ln(x+1) - \ln(1)}{x - 0}.$

   Simplify: $f'(c) = \frac{\ln(x+1) - 0}{x}.$

   Using the derivative $f'(x) = \frac{1}{x+1}$, the MVT guarantees some $c \in (0, x)$ such that: $\frac{1}{c+1} = \frac{\ln(x+1)}{x}.$

3. Prove $\ln(x+1) \leq x$: Since $\frac{1}{c+1} > 0$, this implies $\ln(x+1) \leq x$, as $\frac{\ln(x+1)}{x}$ is always less than or equal to $1$ for $x \geq 0$.

4. Equality Case: For $x = 0$, we directly compute: $\ln(x+1) = \ln(1) = 0.$

   Hence, $\ln(x+1) = x$ when $x = 0$.

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Final Answer:

• (a) $f'(x)$ is odd.
• (b) $\ln(x+1) \leq x$ for $x \geq 0$, with equality at $x = 0$.

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Do you need further explanation on any step? Here are 5 related questions for further practice:

1. How can we prove that the derivative of an odd function is even?
2. What is the geometric interpretation of the Mean Value Theorem?
3. What are examples of real-world applications of even and odd functions?
4. How can we verify $\ln(x+1) \leq x$ graphically?
5. Can the inequality $\ln(x+1) \leq x$ be extended to other domains?

Tip: When solving derivative problems involving symmetry, always use definitions of even and odd functions carefully!