Let $f(x)=\begin{cases}(x-1) \sin \left(\frac{1}{x-1}\right), & \text { if } x \neq 1 \\ 0, & \text { if } x=1\end{cases}$. Then which one of the following is true?

(C) f is differentiable at $x=0$ but not at $x=1$

(A) f is neither differentiable at $x=0$ nor at $x=1$

(B) f is differentiable at $x=0$ and at $x=1$

(D) f is differentiable at $x=1$ but not at $x=0$

JEE Main · 2008 · 27 Apr 2008 · Offline · Continuity and Differentiability

Let

f (x) = {(x - 1) sin (\frac{1}{x - 1}), 0, if x \neq = 1 if x = 1

. Then which one of the following is true?

(A)f is neither differentiable at $x = 0$ nor at $x = 1$
(B)f is differentiable at $x = 0$ and at $x = 1$
(C)f is differentiable at $x = 0$ but not at $x = 1$ ✓ Correct
(D)f is differentiable at $x = 1$ but not at $x = 0$

Answer: (C)

Solution

Away from x = 1 the function is an ordinary product of smooth pieces, so near x = 0 — where the factor 1/(x−1) equals −1 and nothing singular happens — f is perfectly differentiable. The only trouble is at x = 1: there f(1) = 0 and the difference quotient [f(1+h)−f(1)]/h = sin(1/h) swings between −1 and +1 without settling, so no derivative exists. Hence f is differentiable at x = 0 but not at x = 1.

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option

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JEE Main · 2008 · 27 Apr 2008 · Offline · Continuity and Differentiability

Solution

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