Question

The functions
$f\left(x\right)=\{\begin{array}{cc}{e}^x& ifx\le 1\\ mx+b& ifx>1\end{array}$ is continuous and differential at $x=1$. find the values for the constants $m$ and $b$.

Answer 1

$h\to 0f(1-h)=e^{1-h}=ef\left(1\right)=ef(1+h)=m+mh+b=m+bSincef\left(x\right)iscontinuous.m+b=eAlsosincef\left(x\right)isdifferentiable.\frac{f(1+h)-f\left(1\right)}{h}=\frac{f\left(1\right)-f(1-h)}{h}\Rightarrow m+b-e=e-e\Rightarrow m+b=e$