First slide

Differentiability

Question

$Let f (x) = \{\begin{cases} \frac{1}{| x |}, | x | \geq 1 \\ a x^{2} + b, | x | < 1 \end{cases} be continuous and differentiable every where then$

Moderate

A

$a = - 1 / 2, b = 3 / 2$
B

$a = 1 / 2, b = - 3 / 2$
C

$a = 1 / 2, b = 3 / 2$
D

$a = - 1 / 2, b = - 3 / 2$

Solution

$f (x) = \{\begin{cases} \frac{1}{| x |}, | x | \geq 1 \\ a x^{2} + b, | x | < 1 \end{cases} f i s c o n t i n u o u s a t x = 1 a + b = 1 f^{1} (x) = \{\begin{cases} - \frac{1}{x^{2}}, | x | \geq 1 \\ 2 a x, | x | < 1 \end{cases} 2 a = - 1 \Rightarrow a = - \frac{1}{2}, b = \frac{3}{2}$

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