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Continuity and Differenti question bank (Maths)

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Q1.

If $f (x) = \{\begin{cases} x^{p} \cos (\frac{1}{x}), & x \neq 0 \\ 0, & x = 0 \end{cases}$ is differentiable at $x = 0$ , then

(A)

p < 0

(B)

0 < p < 1

(C)

p = 1

(D)

p > 1

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Q2.

Consider a function $g (x) = f (x - 2), \forall x \in R,$ where $f (x) = \{\begin{array}{l} \frac{1}{|x|} & : & |x| \geq 1 \\ a x^{2} + b & : & |x| < 1 \end{array} .$ If $g (x)$ is continuous as well as differentiable for all $x,$ then

(A)

a = \frac{- 1}{2}, b = \frac{3}{2}

(B)

a = \frac{1}{2}, b = \frac{3}{2}

(C)

a = \frac{- 1}{2}, b = \frac{- 3}{2}

(D)

None of these

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