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JEE Main 2008
Exam
JEE Main
Year
2008
Subject
Mathematics
Chapter
Continuity and Differentiability

JEE Main 2008 · Mathematics · Continuity and Differentiability

Let f(x)=\ array ll (x-1) ((1)/(x-1)), & if x ≠ 1 0, & if x=1 array...

Let f(x)=\left\{(x−1)sin⁡(1x−1), if x≠10, if x=1\begin{array}{ll}(x-1) \sin\left(\frac{1}{x-1}\right), & \text{ if } x \neq 1 \\ 0, & \text{ if } x=1\end{array}\right.. Then which one of the following is true?

Options

  1. Aff is neither differentiable at x=0x=0 nor at x=1x=1
  2. Bff is differentiable at x=0x=0 and at x=1x=1
  3. Cf\mathrm{f} is differentiable at x=0\mathrm{x}=0 but not at x=1\mathrm{x}=1
  4. Df\mathrm{f} is differentiable at x=1\mathrm{x}=1 but not at x=0\mathrm{x}=0

Correct answer

A. ff is neither differentiable at x=0x=0 nor at x=1x=1

Step-by-step solution

f′(1)=lim⁡h→0f(1+h)−f(1)hf^{\prime}(1)=\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}\begin{aligned} & \Rightarrow f^{\prime}(1)=\lim _{h \rightarrow 0} \frac{(1+h-1) \sin \left(\frac{1}{1+h-1}\right)-0}{h}=\lim _{h \rightarrow 0} \frac{h}{h} \sin \left(\frac{1}{h}\right) \\ & \Rightarrow f^{\prime}(1)=\lim _{h \rightarrow 0} \sin \left(\frac{1}{h}\right) \end{aligned} ∴f\therefore \mathrm{f} is not differentiable at x=1x=1. Similarly, f′(0)=lim⁡h→0f(h)−f(0)hf^{\prime}(0)=\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h} ⇒f′(0)=lim⁡h→0(h−1)sin⁡(1h−1)−sin⁡(1)h\Rightarrow f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{(h-1) \sin \left(\frac{1}{h-1}\right)-\sin (1)}{h} ⇒f\Rightarrow \mathrm{f} is also not differentiable at x=0\mathrm{x}=0.
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