Menu Close

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) =  { ((x^2 , x≤1)),((2ax+b , x>1)) :}
$$\mathrm{find}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\: \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{function}\:\mathrm{differentiable}\:\mathrm{at}\: \\ $$$$\mathrm{x}=\mathrm{1}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\begin{cases}{\mathrm{x}^{\mathrm{2}} ,\:\mathrm{x}\leqslant\mathrm{1}}\\{\mathrm{2ax}+\mathrm{b}\:,\:\mathrm{x}>\mathrm{1}}\end{cases} \\ $$
Commented by john santu last updated on 23/Apr/20
(1) lim_(x→1^− )  f(x) = lim_(x→1^+ )  f(x)  ⇒ lim_(x→1^− )  (x^2 ) = lim_(x→1^+ )  (2ax+b)  ⇒ 1 = 2a+b   (2) f ′_1^−  (1) = f ′_1^+   (1)  ⇒2(1) = 2a ⇒ a = 1 ∧ b = −1
$$\left(\mathrm{1}\right)\:\underset{{x}\rightarrow\mathrm{1}^{−} } {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\underset{{x}\rightarrow\mathrm{1}^{+} } {\mathrm{lim}}\:{f}\left({x}\right) \\ $$$$\Rightarrow\:\underset{{x}\rightarrow\mathrm{1}^{−} } {\mathrm{lim}}\:\left({x}^{\mathrm{2}} \right)\:=\:\underset{{x}\rightarrow\mathrm{1}^{+} } {\mathrm{lim}}\:\left(\mathrm{2}{ax}+{b}\right) \\ $$$$\Rightarrow\:\mathrm{1}\:=\:\mathrm{2}{a}+{b}\: \\ $$$$\left(\mathrm{2}\right)\:{f}\:'_{\mathrm{1}^{−} } \left(\mathrm{1}\right)\:=\:{f}\:'_{\mathrm{1}^{+} } \:\left(\mathrm{1}\right) \\ $$$$\Rightarrow\mathrm{2}\left(\mathrm{1}\right)\:=\:\mathrm{2}{a}\:\Rightarrow\:{a}\:=\:\mathrm{1}\:\wedge\:{b}\:=\:−\mathrm{1} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *