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7-123.

Determine the values of a and b so that the function defined below is continuous and differentiable. Homework Help ✎

$f ( x ) = \left\{ \begin{array} { l l } { a e ^ { x } } & { \text { for } x \leq 1 } \\ { \operatorname { ln } ( x ) + b } & { \text { for } x > 1 } \end{array} \right.$

$\text{Continuous means }\lim_{x\rightarrow 1^{-}}f(x)=\lim_{x\rightarrow 1^{+}}f(x)=f(1).$

Differential means $\lim\limits_{x\rightarrow 1^{-}}f'(x)=\lim\limits_{x\rightarrow 1^{+}}f'(x)=f(1)$, but only if it's continuous as well.

Write and solve a system of equations, each evaluated at $x = 1$. The first equation will equate the two pieces of $f(x)$.
The second equation will equate the two pieces of $f^\prime(x)$.