Question

The function \[ f(x)= \begin{cases} 2x^2-1, & \text{if } 1 \leq x \leq 4 \\ 151-30x, & \text{if } 4 < x \leq 5 \end{cases} \] is not suitable to apply Rolle's theorem since:

• A. \( f(x) \) is not continuous on \( [1, 5] \)
• B. \( f(1) \neq f(5) \)
• C. \( f(x) \) is continuous only at \( x = 4 \)
• D. \( f(x) \) is not differentiable in \( (4, 5) \)
• E. \( f(x) \) is not differentiable at \( x = 4 \)

Hint:

Even if a piecewise function "connects" (is continuous), it can still have a "sharp corner" where the slopes don't match. This corner makes it non-differentiable.

Answer 1

Answer: (E)

Concept: Rolle's Theorem requires three conditions for a function on \( [a, b] \): 1. Continuity on \( [a, b] \). 2. Differentiability on \( (a, b) \). 3. \( f(a) = f(b) \).

Step 1: Check continuity at \( x = 4 \).
LHL: \( 2(4)^2 - 1 = 32 - 1 = 31 \). RHL: \( 151 - 30(4) = 151 - 120 = 31 \). Since LHL = RHL, the function is continuous at \( x = 4 \).

Step 2: Check the endpoint values.
\( f(1) = 2(1)^2 - 1 = 1 \). \( f(5) = 151 - 30(5) = 151 - 150 = 1 \). So \( f(1) = f(5) \).

Step 3: Check differentiability at \( x = 4 \).
Left-hand derivative (LHD): \( \frac{d}{dx}(2x^2 - 1) = 4x \). At \( x=4 \), LHD = 16. Right-hand derivative (RHD): \( \frac{d}{dx}(151 - 30x) = -30 \). Since LHD \( \neq \) RHD (\( 16 \neq -30 \)), the function is not differentiable at \( x = 4 \).