Chapter 12.5, Problem 57ES

These exercises are concerned with the problem of creating a single smooth curve by piecing together two separate smooth curves. If two smooth curves $C_1\text{ and }{C}_2$ are joined at a point P to form a curve C, then we will say that $C_1\text{ and }{C}_2$ make a smooth transition at P if the curvature of C is continuous at P.

Assume that f is a function for which $f^{‴}\left(x\right)$ is defined for all $x\le 0.$ Explain why it is always possible to find numbers a, b, and c such that there is a smooth transition at $x=0$ from the curve $y=f\left(x\right),x\le 0,$ to the parabola $y=a{x}^2+bx+c.$