If $f: R ightarrow R$ defined by $$ f(x)=\left\{\begin{array}{cc} a^2 \cos ^2 x+b^2 \sin ^2 x, & x \leq 0 \ e^{a x+b}, & x0 \end{array} ight. $$ is a continuous function, then

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If $f: R ightarrow R$ defined by $$ f(x)=\left\{\begin{array}{cc} a^2 \cos ^2 x+b^2 \sin ^2 x, & x \leq 0 \ e^{a x+b}, & x>0 \end{array}ight. $$ is a continuous function, then

Aicharya · Accepted Answer

We have, $ \begin{aligned} & \lim _{h ightarrow 0} f(0+h)=\lim _{h ightarrow 0} e^{a h+b}=e^b \ & 	ext { and } \lim _{h ightarrow 0} f(0-h)=\lim _{h ightarrow 0} a^2 \cos ^2 h \ &+b^2 \sin ^2 h=a^2 0 \ & 	ext { Thus, } \quad e^b=a^2 \ & b=2 \log |a| \end{aligned} $

If $f: R \rightarrow R$ defined by $$ f(x)=\left\{\begin{array}{cc} a^2 \cos ^2 x+b^2 \sin ^2 x, & x \leq 0…

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