Question

Let \(f(x) = \begin{cases} 5x^2 + ax + 16, & \text{if } x < 2 \\ x^2, & \text{if } x \geq 2 \end{cases}\). If \(f\) is differentiable at \(x = 2\), then the value of \(a\) is equal to

• A. 16
• B. -18
• C. 32
• D. -32
• E. -16

Hint:

Differentiability implies continuity. Always check continuity first to find possible values.

Answer 1

Answer: (E)

Step 1: Concept:
• For differentiability at \(x = 2\):
• Function must be continuous at \(x = 2\)
• Left-hand derivative (LHD) = Right-hand derivative (RHD)

Step 2: Detailed Explanation:
• Continuity condition: \[ \lim_{x \to 2^-} (5x^2 + ax + 16) = f(2) = 4 \]
• Substitute \(x = 2\): \[ 5(4) + 2a + 16 = 4 \Rightarrow 20 + 2a + 16 = 4 \Rightarrow 36 + 2a = 4 \Rightarrow 2a = -32 \Rightarrow a = -16 \]
• Check differentiability:
• Left-hand derivative (LHD): \[ f'(x) = 10x + a \quad (x<2) \] \[ \text{LHD at } x=2 = 20 + a = 20 - 16 = 4 \]
• Right-hand derivative (RHD): \[ f'(x) = 2x \quad (x>2) \] \[ \text{RHD at } x=2 = 4 \]
• Since LHD = RHD, function is differentiable.

Step 3: Final Answer:
• \[ a = -16 \]