Question

If $f\left(x\right)=\{\begin{array}{c}{a}^2{\cos }^{2}x+b^2{\sin }^{2}x,x\le 0\\ e^{ax+b},x>0\end{array}$ $f\left(x\right)$ is continuous at $x=0$ then

• A. $2log|a|=b$
• B. $2log|b|=e$
• C. $loga=2log|b|$
• D. $a=b$

Answer 1

The correct option is A $2log|a|=b$

Given the function $f\left(x\right)$ is continuous at $x=0$, then

$\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}f\left(x\right)$

or, $\underset{x\to 0^{-}}{lim}(a^2{\cos }^{2}x+b^2{\sin }^{2}x)=\underset{x\to 0^{+}}{lim}\left(e^{ax+b}\right)$

or, $a^2=e^b$

or, $b=2\log |a|$.