a-Show-that-f-x-x-is-derivable-at-all-points-x-0-gt-0-and-that-f-x-0-1-2x-0-b-Show-that-the-function-f-x-x-continuous-at-x-0-0-is-not-derivable-at-x-0-0-

Question Number 95638 by Ar Brandon last updated on 26/May/20

a ∖ Show that f (x) = \sqrt{x} is derivable at all points x_{0} > 0

and that f^{'} (x_{0}) = \frac{1}{{2 x}_{0}}

b ∖ Show that the function f (x) = \sqrt{x} (continuous at x_{0} = 0)

is not derivable at x_{0} = 0

Answered by john santu last updated on 26/May/20

(b) f (x) = \sqrt{x} continuous at x_{0} = 0

(i) \lim_{x \to 0^{-}} \sqrt{x} = \lim_{x \to 0^{+}} \sqrt{x} = 0

(ii) f (0) = \sqrt{0} = 0

since \lim_{x \to 0} \sqrt{x} = f (0), hence f (x) = \sqrt{x}

continuous at x_{0} = 0