Discussion
Let
\[ f(x) = \begin{cases} 5x^{2} + ax + 16, & \text{if } x \lt 2 \\ x^{2}, & if x \ge 2 \end{cases} \]
is differentiable at \(x = 2\) , then the value of a is equal to
Answer: E
\(a =\) \(-16\).
Differentiability at 2 needs the two pieces to have the same value and the same slope there:
\[ \begin{aligned} \text{value: } 20 + 2a + 16 &= 4 \;\Rightarrow\; a = -16 \\ \text{slope: } 10(2) + a &= 2(2) \;\Rightarrow\; a = -16 \end{aligned} \]
Both conditions agree, so \(a = -16\). \(16\) and \(\pm 32\) come from sign or factor-of-2 slips, and \(-18\) satisfies neither condition.
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