Question

Let S = {t ∈ R : f(x) = |x – π| (e^|x| – 1)sin |x| is not differentiable at t}. Then the set S is equal to ______.

Options

• {0}

• {π}

• {0, π}

• `phi` (an empty set)

Answer 1

Let S = {t ∈ R : f(x) = |x – π| (e^|x| – 1)sin |x| is not differentiable at t}. Then the set S is equal to `underlinebb(phi ("an empty set"))`.

Explanation:

f(x) = |x – π| (e^|x| – 1)sin |x|

Check differentiability of f(x) at x = π and x = 0

at x = π:

R.H.D. = `lim_(h rightarrow 0) (|π + h - π|(e^(|x + h|) - 1) sin|π + h| - 0)/h` = 0

L.H.D. = `lim_(h rightarrow 0) (|π - h - π|(e^(|x - h|) - 1) sin|π - h| - 0)/(-h)` = 0

∵ R.H.D. = L.H.D.

Therefore, the function is differentiable at x = π

at x = 0:

R.H.D. = `lim_(h rightarrow 0) (|h - π|(e^(|h|) - 1) sin|h| - 0)/h` = 0

L.H.D. = `lim_(h rightarrow 0) (|- h - π|(e^(|-h|) - 1) sin|-h| - 0)/(-h)` = 0

∴ R.H.D. = L.H.D.

Therefore, the function is differentiable at x = 0.

Since, the function f(x) is differentiable at all the points including π and 0.

i.e., f(x) is everywhere differentiable.

Therefore, there is no element in the set S.

`\implies` S = `phi` (an empty set)