Limits, Continuity & Differentiability
Differentiability
Grade 12
Question:
<p>If \( f(x) = \begin{cases} ax + b, & x \le -1 \\ ax^3 + x + 2b, & x > -1 \end{cases} \) is differentiable for all \( x \in \mathbb{R} \), then find \( b - a = \) __________.</p>
Step-by-Step Solution
Key Concept: For differentiability at x = -1, both continuity and equal left/right derivatives must hold simultaneously. Use these two conditions to set up a system of equations in a and b.
<p><strong>Step 1: Continuity at x = -1</strong></p><p>Left limit: f(-1) = a(-1) + b = -a + b</p><p>Right limit: lim(x→-1⁺) [ax³ + x + 2b] = a(-1)³ + (-1) + 2b = -a - 1 + 2b</p><p>For continuity: -a + b = -a - 1 + 2b</p><p>This gives: <strong>b = 1</strong> ... (Equation 1)</p><p><strong>Step 2: Differentiability at x = -1 (equal derivatives)</strong></p><p>Left derivative: f'(-1⁻) = a</p><p>Right derivative: f'(-1⁺) = 3ax² + 1|ₓ₌₋₁ = 3a(1) + 1 = 3a + 1</p><p>For differentiability: a = 3a + 1</p><p>This gives: -2a = 1, so <strong>a = -1/2</strong> ... (Equation 2)</p><p><strong>Step 3: Calculate b - a</strong></p><p>b - a = 1 - (-1/2) = 1 + 1/2 = <strong>3/2</strong></p><p>∴ Answer: <strong>3/2</strong></p>
Correct Answer: 3