Equation of Parabola in Rosenbrock function

Question

The Rosenbrock function is a popular benchmark for optimization, its general form is $$ f(x, y) = (a - x)^2 + b(y - x^2)^2 $$ where typically $a=1$ and $b=100$. The function looks like this

Is it possible to find the equation of the parabola that would fit within its contours? Here's a rough drawing.

Answer 1

It's just $y=x^2$.

For any fixed $x$ value (i.e. vertical line), the bottom of the trough, where $f$ is minimal along that line, occurs where $\partial f/\partial y = 0$. And $\partial f/\partial y = 2b(y-x^2)$.