Question

Check for differentiability of the function f defined by f(x) = |x − 5|, at the point x = 5.

Answer 1

Given f(x) = |x − 5|

∴ f(x) = `{((x - 5), x ≥ 5),(-(x - 5), x < 5):}`

Here, LHD = `lim_(h->0) (f(x - h) - f (x))/-h`

= `lim_(h->0) (f(5 - h) - f (5))/-h`

= `lim_(h->0) ((-5 - h - 5) - 0)/-h`

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

and RHD = `lim_(h->0) (f(5 + h) - f (5))/h`

= `lim_(h->0) ((5 + h - 5) - 0)/h`

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

= 1

∵ LHD ≠ RHD Hence f(x) is not differentiable.