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: Use the given conditions. The problem gives that:
\( f(x) \) 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: Interpret the condition on the derivative.
The condition \( |f'(x)| \leq \frac{1}{5} \) implies that the slope of the tangent to the curve at any point is at most \( \frac{1}{5} \). Therefore, the function \( f(x) \) can change at a rate no faster than \( \frac{1}{5} \) as \( x \) increases.
Step 3: Consider the simplest linear function.
Given that \( f(0) = 0 \) and the derivative bound, the most natural function that satisfies these conditions is a linear function of the form \( f(x) = \frac{x}{5} \). This is because the derivative of \( f(x) = \frac{x}{5} \) is: \[ f'(x) = \frac{1}{5}, \] which satisfies \( |f'(x)| \leq \frac{1}{5} \) for all \( x \in (0, 5) \), and the function also passes through the origin.
Step 4: Conclude the solution.
Thus, the function \( f(x) = \frac{x}{5} \) satisfies all the given conditions, and therefore, the correct answer is: \[ f(x) = \frac{x}{5}. \]