Let f(x)=\ array c (x-1) (1)/(x-1) if x ≠ 1 0 if x=1 array . Then which one of the following is true?

Correct answer: B. f is differentiable at x=0 but not at x=1 We have; f(x)=\ array ll (x-1) ((1)/(x-1)) & if x ≠ 1 0 & if x=1 array .R f^ (1)= _ h arrow 0 (f(1+h)-f(1))/(h)= _ h arrow 0 (h (1)/(h)-0)/(h)= _ h arrow 0 (1)/(h) which does not exist. f is not differentiable at x=1 Also .f^ (0)= (1)/((x-1))- x-1 (x-1)^ 2 ((1)/(x-1))]_ x=0 =- 1+ 1 f is differentiable at x=0

BITSAT 2014

Exam: BITSAT
Year: 2014
Subject: Mathematics
Chapter: Continuity and Differentiability

BITSAT 2014 · Mathematics · Continuity and Differentiability

Let f(x)=\ array c (x-1) (1)/(x-1) if x ≠ 1 0 if x=1 array . Then...

Let

f(x)=\left\{

\begin{array}{c}(x-1) \sin\frac{1}{x-1} \text{ if } x \neq 1 \\ 0 \quad \text{ if } x=1\end{array}

\right.

Then which one of the following is true?

Options

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

Correct answer

B. $f$ is differentiable at $x=0$ but not at $x=1$

Step-by-step solution

We have;

f(x)=\left\{

\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.R f^{\prime}(1)=\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}=\lim _{h \rightarrow 0} \frac{h \sin \frac{1}{h}-0}{h}=\lim _{h \rightarrow 0} \sin \frac{1}{h}

which does not exist.

\therefore f

is not differentiable at

x=1

Also \left.f^{\prime}(0)=\sin \frac{1}{(x-1)}-\frac{x-1}{(x-1)^{2}} \cos \left(\frac{1}{x-1}\right)\right]_{x=0}=-\sin 1+\cos 1\therefore f

is differentiable at

x=0

Practice Continuity and Differentiability on Quantrex Academy →

More from Continuity and Differentiability

Let f be the function defined by f(x)=\ array cc (x^2-1)/(x^2-2|x-1|-1), & x ≠ 1 (1)/(2)… BITSAT 2024 If f(x)=\ array cc (x^2 ( x))/( (1+x)), & x ≠ 0 0, & x=0 array . , then at x=0, f(x) is BITSAT 2024 f(x)=\ array cc 4 & -∞ < x < - 5 x^2-1 & - 5 ≤ x ≤ 5 4 & 5 ≤ x < ∞ array . If k is the… BITSAT 2024 Let f be the function defined by f(x)=\ array ll x^ 2 -1 x^ 2 -2|x-1|-1 , & x ≠ 1 1 / 2… BITSAT 2020 Let f(x) = \ array ll x^ p (1)/(x), x ≠ 0 array cases 0, & x = 0 cases ) then f (x) is… BITSAT 2018 If f(x) = \ cases e^ x , & x ≤ 0 |1 - x|, & x > 0 cases ), then BITSAT 2017 The value of f ( 0 ) , so that the function f(x) = 2x - ^ -1 x 2x + ^ -1 x is continuous… BITSAT 2017 Consider the greatest integer function, defined by f ( x ) = [ x ] , 0 ≤ x < 2 . Then BITSAT 2017 The number of points at which the function f(x)= 1 |x| is discontinuous is: BITSAT 2016 If f(x)=\ array cl x x (1+x^ 2 ) & , x ≠ 0 0 & , x=0 array . then f(x) is BITSAT 2016 If a function f(x) is given by f(x)=(x)/(1+x)+(x)/((x+1)(2 x+1))+(x)/((2 x+1)(3 x+1))+…… BITSAT 2015 Let f: R arrow R be a function such that f(x+y)=f(x)+f(y), x, y ∈ ℝ If f(x) is… BITSAT 2015 Let f(x)=\ array l 1- ^ 3 x 3 ^ 2 x , x (π)/(2) array . If f(x) is continuous at… BITSAT 2014 If the function f(x)= cases A x-B, & x ≤ 1 3 x & , 1< x< 2 B x^ 2 -A, & x ≥ 2 cases is… BITSAT 2012 The function f(x)=(x-1) |ℓ n x| is at x=1) BITSAT 2011 The value of | array ccc 1 & 2 & 3 -4 & 3 & 6 2 & -7 & 9 array | is BITSAT 2010 Full Continuity and Differentiability list Continuity and Differentiability JEE Maths PYQs JEE Mathematics PYQs All BITSAT PYQs