Question

Consider the function \( f(t) = (\max(0, t))^2 \) for \( -\inftyLt;tLt;\infty \), where \( \max(a, b) \) denotes the maximum of \( a \) and \( b \). Which of the following statements is/are true?

• A. \( f(t) \text{ is not differentiable.} \)
• B. \( f(t) \text{ is differentiable and its derivative is continuous.} \)
• C. \( f(t) \text{ is differentiable but its derivative is not continuous.} \)
• D. \( f(t) \text{ and its derivative are differentiable.} \)

Hint:

For piecewise functions, always check: 1. Differentiability at the junction points using left-hand and right-hand derivatives. 2. Continuity of the derivative \( f'(t) \) to determine smoothness.

Answer 1

The Correct Option is B

Step 1: Understanding the function \( f(t) \). The given function is defined as \( f(t) = (\max(0, t))^2 \). Breaking this into cases: - For \( t \geq 0 \), \( \max(0, t) = t \), so \( f(t) = t^2 \). - For \( tLt;0 \), \( \max(0, t) = 0 \), so \( f(t) = 0 \). Step 2: Analyzing differentiability. - For \( t>0 \): \( f(t) = t^2 \), and its derivative \( f'(t) = 2t \). - For \( tLt;0 \): \( f(t) = 0 \), so \( f'(t) = 0 \). - At \( t = 0 \): The left-hand derivative \( \lim_{t \to 0^-} f'(t) = 0 \) and the right-hand derivative \( \lim_{t \to 0^+} f'(t) = 0 \). Thus, \( f(t) \) is differentiable at \( t = 0 \). Step 3: Checking the continuity of \( f'(t) \). While \( f'(t) \) exists at all points, its value changes abruptly from \( 0 \) for \( tLt;0 \) to \( 2t \) for \( t>0 \). This discontinuity at \( t = 0 \) makes \( f'(t) \) not continuous. Step 4: Final Answer. The function \( f(t) \) is differentiable everywhere, but its derivative \( f'(t) \) and its derivative is continuous at \( t = 0 \).