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

$$\mathrm{find}\mathrm{the}\mathrm{values}\mathrm{of}\mathrm{a}\mathrm{and}\mathrm{b}$$ $$\mathrm{such}\mathrm{that}\mathrm{the}\mathrm{following}$$ $$\mathrm{function}\mathrm{differentiable}\mathrm{at}$$ $$\mathrm{x}=1\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{l}{\mathrm{x}}^2,\mathrm{x}⩽1\\ 2\mathrm{ax}+\mathrm{b},\mathrm{x}>1\end{array}$$

Commented byjohn santu last updated on 23/Apr/20

$$\left(1\right)\underset{x\to 1^{-}}{\lim }f\left(x\right)=\underset{x\to 1^{+}}{\lim }f\left(x\right)$$ $$\Rightarrow \underset{x\to 1^{-}}{\lim }\left(x^2\right)=\underset{x\to 1^{+}}{\lim }(2ax+b)$$ $$\Rightarrow 1=2a+b$$ $$\left(2\right)f{}_{1^{-}}^{\prime }\left(1\right)=f{}_{1^{+}}^{\prime }\left(1\right)$$ $$\Rightarrow 2\left(1\right)=2a\Rightarrow a=1\wedge b=-1$$

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