Question

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

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

Hint:

If derivative is bounded: Use MVT. Bound function growth linearly.

Answer 1

The Correct Option is A

Concept: Use Mean Value Theorem: \[ f(x) - f(0) = f'(c)x \] for some \( c \in (0,x) \). Step 1: Apply MVT. Since \( f(0)=0 \): \[ f(x) = f'(c)x \] Step 2: Use derivative bound.} \[ |f(x)| = |f'(c)||x| \le \frac{1}{5} \cdot 5 = 1 \] Step 3: Conclusion.} \[ |f(x)| \le 1 \quad \forall x \in [0,5] \]