Question

If f(x) = `{(2x - 3",", -3 ≤ x ≤ -2), (x + 1",", -2 ≤ x ≤ 0):}`

Check the differentiability of f(x) at x = -2.

Answer 1

f(x) = `{(2x - 3",", -3 ≤ x ≤ -2), (x + 1",", -2 ≤ x ≤ 0):}`

LHD at -2

= `lim_(h -> 0)(f(a - h) - f(a))/-h`

= `lim_(h -> 0)(f(-2 - h) - f(-2))/-h`

= `lim_(h -> 0)([2(-2 - h) - 3] - [2(-2) - 3])/-h`

= `lim_(h -> 0)(-4 - 2h - 3 + 4 + 3)/-h`

= `lim_(h -> 0)(-2h)/-h`

= 2

RHD = `lim_(h -> 0)(f(a + h) - f(a))/h`

= `lim_(h -> 0)((-2 + h + 1) - (-2 + 1))/h`

= `lim_(h -> 0)((-1 + h) - (-1))/h`

= `lim_(h -> 0)(h)/h`

= 1

∴ LHD ≠  RHD

∴ Not differentiable at x = −2