devarshi-dt-logo

Question:

Let S = {t ∈ R: f(x) = |x − π| ⋅ (e|x|)sin|x| is not differentiable at t}. Then the set S is equal to

∅(an empty set)

π

0, π

0

Solution:

We have to check the differentiability of f(x) at x = 0 and π.

At x = 0
RHL = limh→0 [f(x + h) − f(x)]/h = limh→0 [f(h) − f(0)]/h = |h − π|(eh)sinh/h = limh→0 π(eh)sinh/h = limh→0 π × 0 × sinh/h = 0 × 1 = 0
Since f(0) = 0
limh→0 sinh/h = 1
For Left-hand limit at x = 0
LHL = limh→0 [f(x) − f(x − h)]/−h = limh→0 [f(0) − f(−h)]/−h = limh→0 [0 − (|−h − π|(eh)sin|−h|)]/−h = limh→0 π(eh)sinh/−h = limh→0 π × 0 × sinh/−h = 0 × 1 = 0
LHL = RHL ⇒ f(x) is differentiable at x = 0

  1. At x = π
    RHL = limh→0 [f(x + h) − f(x)]/h = limh→0 [f(π + h) − f(π)]/h = |π + h − π|(eπ)sin(π + h)/h = limh→0 h(eπ)sin(π + h)/h = limh→0 h × (eπ) × sinπ/h = 0 × 1 = 0
    Since f(π) = 0
    For Left-hand limit at x = π
    LHL = limh→0 [f(x) − f(x − h)]/−h = limh→0 [f(π) − f(π − h)]/−h = limh→0 [0 − (|π − h − π|(eπ−h)sin|π − h|)]/−h = limh→0 h(eπ)sin(π + h)/h = limh→0 h × (eπ) × sinπ/h = 0
    LHL = RHL ⇒ f(x) is differentiable at x = π
    Set S is an empty set.