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.