isi-entrance 2012 Q7

isi-entrance · India · solved Differentiating Transcendental Functions Regularity and smoothness of transcendental functions

Let $f(x) = e^{-1/x}$ for $x > 0$ and $f(x) = 0$ for $x \leq 0$. Which of the following is true?
(A) $f$ is not differentiable at $x = 0$
(B) $f$ is differentiable at $x = 0$ but $f'$ is not differentiable at $x = 0$
(C) $f$ is differentiable at $x = 0$ and $f'$ is differentiable at $x = 0$
(D) $f$ is differentiable everywhere and $f'$ is also differentiable everywhere

$f$ is differentiable at $x=0$ since $\lim_{x\to 0+}\frac{f(x)-f(0)}{x} = \lim_{x\to 0+}\frac{e^{-1/x}}{x} = 0$ and the left limit is also $0$. Also $f'(x) = e^{-1/x}/x^2$ for $x>0$ and $0$ for $x\leq 0$, and $f'$ is differentiable at $0$ as well. (D)

Let $f(x) = e^{-1/x}$ for $x > 0$ and $f(x) = 0$ for $x \leq 0$. Which of the following is true?

(A) $f$ is not differentiable at $x = 0$

(B) $f$ is differentiable at $x = 0$ but $f'$ is not differentiable at $x = 0$

(C) $f$ is differentiable at $x = 0$ and $f'$ is differentiable at $x = 0$

(D) $f$ is differentiable everywhere and $f'$ is also differentiable everywhere

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