Chapter 12.5, Problem 56ES

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.

Find a, b, and c so that there is a smooth transition at $x=0$ from the curve $y=e^x\text{for }x\le 0$ to the parabola $y=a{x}^2+bx+c\text{ for }x>0.$