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Question: Let ƒ(x) = \(f(x) = \frac{\sin(e^{x - 2} - 1)}{\log(x - 1)},\) then –...

Let ƒ(x) = f(x)=sin⁡(ex−2−1)log⁡(x−1),f(x) = \frac{\sin(e^{x - 2} - 1)}{\log(x - 1)}, then –

A

ƒ(x) is continuous everywhere

B

ƒ(x) is not continuous at x = 2

C

ƒ(x) is not differentiable at exactly three points

D

None of these

Answer

ƒ(x) is not continuous at x = 2

Explanation

Solution

If x < 2,

f(x) = ∫0x{5+∣1−t∣}dt=∫01(5+1−t)dt+∫1x(5+t−1)dt\int _ { 0 } ^ { \mathrm { x } } \{ 5 + | 1 - \mathrm { t } | \} \mathrm { dt } = \int _ { 0 } ^ { 1 } ( 5 + 1 - \mathrm { t } ) \mathrm { dt } + \int _ { 1 } ^ { \mathrm { x } } ( 5 + \mathrm { t } - 1 ) \mathrm { dt }

= [6t−t22]01\left[ 6 \mathrm { t } - \frac { \mathrm { t } ^ { 2 } } { 2 } \right] _ { 0 } ^ { 1 }+ = 1 + 4x + x22\frac { x ^ { 2 } } { 2 }

So, f(x) = {1+4x+x22, if x<25x+1, if x≥2\left\{ \begin{array} { c c } 1 + 4 x + \frac { x ^ { 2 } } { 2 } , & \text { if } x < 2 \\ 5 x + 1 , & \text { if } x \geq 2 \end{array} \right.

Clearly f(x) is continuous every where including x = 2

Again f′(x) =

f′(x) is not continuous at x = 2

∴ f(x) is not differentiable only at x = 2.