Answer
\[
1 / 2=a, \quad 1=c, \quad 1=b
\]
Work Step by Step
Given
\[
y=\left\{\begin{array}{ll}
e^{x} & x \leq 0 \\
a x^{2}+b x+c & x>0
\end{array}\right.
\]
Since
\[
\frac{\left|y^{\prime \prime}\right|}{\left[y^{\prime 2}+1\right]^{3 / 2}}=\kappa(x)
\]
The transition will be smooth if the values of $y$ are equal, the values of $y^{\prime}$ are equal, and the values of $y^{\prime \prime}$ are equal at $0=x$.
We get
\[
\begin{array}{l}
2 a x+b=y^{\prime} \\
2 a=y^{\prime \prime}
\end{array}
\]
Thus:
\[
1 / 2=a, \quad 1=c, \quad 1=b
\]
