Question

Let \( f(x) \) be continuous on \( [0, 5] \) and differentiable in \( (0, 5) \). If \( f(0) = 0 \) and \( |f'(x)| \leq \frac{1}{5} \) for all \( x \) in \( (0, 5) \), then for all \( x \) in \( [0, 5] \):

• A. \( |f(x)| \leq 1 \)
• B. \( |f(x)| \leq \frac{1}{5} \)
• C. \( f(x) = \frac{x}{5} \)
• D. \( |f(x)| \geq 1 \)

Hint:

When given a bound on the derivative of a function, the simplest function that meets the condition is often a linear function with the given slope. In this case, \( f(x) = \frac{x}{5} \) satisfies the condition \( |f'(x)| \leq \frac{1}{5} \) and passes through the origin.

Answer 1

The Correct Option is C

Step 1: State the given conditions.
The problem provides:
\( f(x) \) is continuous on \( [0, 5] \) and differentiable in \( (0, 5) \),
\( f(0) = 0 \),
\( |f'(x)| \leq \frac{1}{5} \) for all \( x \in (0, 5) \).

Step 2: Explain the derivative condition.
The condition \( |f'(x)| \leq \frac{1}{5} \) means the tangent slope's magnitude is at most \( \frac{1}{5} \). Thus, \( f(x) \) changes no faster than \( \frac{1}{5} \) as \( x \) increases.

Step 3: Consider a simple linear function.
Given \( f(0) = 0 \) and the derivative bound, consider \( f(x) = \frac{x}{5} \). Its derivative is:
\[\nf'(x) = \frac{1}{5},\n\]which satisfies \( |f'(x)| \leq \frac{1}{5} \) for all \( x \in (0, 5) \), and passes through the origin.

Step 4: Present the solution.
Therefore, \( f(x) = \frac{x}{5} \) satisfies all conditions, so the answer is:
\[\nf(x) = \frac{x}{5}.\n\]