Applications of Derivatives
Maxima and Minima
Grade 12
Question:
<p><strong>527.</strong> Let \( f(x) = \begin{cases} -x^3 + a, & 0 \leq x < 1 \\ x, & 1 \leq x \leq 3 \end{cases} \)</p><p>Which of the following statements is/are correct?</p>
<p>(a) if \( f(x) \) has an absolute minimum at \( x = 1 \), then minimum positive integral value of \( a \) is 2.</p>
<p>(b) if \( f(x) \) has an absolute minimum at \( x = 1 \), then minimum positive integral value of \( a \) is 3.</p>
<p>(c) if \( f(x) \) has an absolute maximum at \( x = 3 \), then maximum positive integral value of \( a \) is 3.</p>
<p>(d) if \( f(x) \) has an absolute maximum at \( x = 0 \), then minimum positive integral value of \( a \) is 3.</p>
Step-by-Step Solution
Key Concept: For a piecewise function to be differentiable at the junction point, both continuity AND equality of left/right derivatives must hold simultaneously. This creates a system of equations that constrains the parameters.
<p><strong>Step 1: Continuity at x = 0</strong></p><p>From left: f(0⁻) = -0³ + a = a</p><p>From right: f(0⁺) = b|0-1| = b</p><p>For continuity: <strong>a = b</strong></p><p><strong>Step 2: Derivative from left (x < 0)</strong></p><p>f'(x) = -3x², so f'(0⁻) = 0</p><p><strong>Step 3: Derivative from right (0 < x < 1)</strong></p><p>f(x) = b(1-x), so f'(x) = -b, thus f'(0⁺) = -b</p><p><strong>Step 4: Differentiability at x = 0</strong></p><p>For differentiability: f'(0⁻) = f'(0⁺)</p><p>0 = -b, so <strong>b = 0</strong></p><p>Therefore: <strong>a = b = 0</strong></p><p><strong>Step 5: Check each statement with a = b = 0</strong></p><p><strong>A:</strong> f'(0) = 0 (exists) ✓ but f'(1) DNE from the piecewise definition at boundary</p><p><strong>B:</strong> f is continuous on [0,1] ✓ (all pieces continuous, joined smoothly)</p><p><strong>C:</strong> f is differentiable on (0,1) ✓ (both pieces differentiable in their interiors)</p><p><strong>D:</strong> f is differentiable on [0,1) ✓ (differentiable at 0 and throughout (0,1))</p><p>∴ Answer: <strong>B, C, D</strong></p>
Correct Answer: B,C,D