Answer
See explanation.
Work Step by Step
Given
\[
y=\left\{\begin{array}{ll}
f(x) & x \leq 0 \\
a x^{2}+b x+c & x>0
\end{array}\right.
\]
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
\[
\begin{aligned}
2 a x+b=y^{\prime} & \\
2 a=y^{\prime \prime} &
\end{aligned}
\]
At $0=x$ and equate $y, y^{\prime}, y^{\prime \prime},$ we get
\[
f^{\prime}(0)=b, \quad f(0)=c, \quad f^{\prime \prime}(0) / 2=a
\]
It's clear that we don't need $f^{\prime \prime \prime}(x)$ to exist for all $x \leq 0 ;$ it suffices to have $f^{\prime \prime}(x)$ continuous.
