find-the-values-of-a-and-b-such-that-the-following-function-differentiable-at-x-1-f-x-x-2-x-1-2ax-b-x-gt-1-

Question Number 90383 by jagoll last updated on 23/Apr/20

find the values of a and b

such that the following

function differentiable at

x = 1 f (x) = {\begin{cases} x^{2}, x ⩽ 1 \\ 2 ax + b, x > 1 \end{cases}

Commented by john santu last updated on 23/Apr/20

(1) \lim_{x \to 1^{-}} f (x) = \lim_{x \to 1^{+}} f (x)

\Rightarrow \lim_{x \to 1^{-}} (x^{2}) = \lim_{x \to 1^{+}} (2 a x + b)

\Rightarrow 1 = 2 a + b

(2) f_{1^{-}}^{'} (1) = f_{1^{+}}^{'} (1)

\Rightarrow 2 (1) = 2 a \Rightarrow a = 1 \land b = - 1