Q41 / 93 · NTA Abhyas (JEE Main) 2020 · Mathematics · Continuity and Differentiability

Consider the function fx=sinx-4.tan-11x-4x≠40x=4, then

  • A
    f(x) is continuous and differentiable at x=4
  • Correct answer
    B
    fx is continuous but non differentiable at x=4
  • C
    fx is discontinuous but differentiable at x=4
  • D
    fx is discontinuous and non differentiable at x=4

Solution

Let us shift the graph of fx in the direction of the negative x-axis by 4 units
& let gx=fx+4
∵gx at x=0 will have the same behaviour as of fx at x=4
gx=sin⁡x. tan-11x0 x≠0  x=0
Continuity at x=0,
limx→0-gx=limx→0-sin⁡x.tan-11x=0×-π/2=0
limx→0+gx=limx→0+sin x.tan-11x=0×π/2=0
∴limx→0+gx=g0 ∴gx is continuous at x=0
⇒fx is continuous at x=4
Differentiability at x=0
g'0−=limh→0+g0−h−g0−h=limh→0+sin−htan−1−1h−0−h
⇒limh→0+-sin hhtan-11h=-1.π2=-π2
g'0+=limh→0+g0+h−g0h=limh→0+sin h.tan−11h−0h
⇒limh→0+sin hhtan-11h=π2
Clearly, g'0−≠g'0+
∴gx is non-differentiable at x=0
⇒fx is non-differentiable at x=4

Hence, the correct answer is option: 'B'.

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