First slide

Differentiability

Question

Let $f (x) = \{\begin{cases} a x^{2} + 1 f o r x > 1 \\ x + a f o r x \leq 1 \end{cases}$ then $f (x)$ is derivable at $x = 1$ , if

Moderate

A

$a = 0$
B

$a = \frac{1}{2}$
C

$a = 1$
D

$a = 2$

Solution

$f (x) = \{\begin{cases} a x^{2} + 1 f o r x > 1 \\ x + a f o r x \leq 1 \end{cases} f (x) = \{\begin{cases} 2 a x f o r x > 1 \\ 1 f o r x \leq 1 \end{cases} 2 a = 1 \Rightarrow a = \frac{1}{2}$

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