Question

Optimisation problem:

What is the minimum vertical distance between the parabolas y = x 2 + 1 and y = x - x^2 ?

Answer 1

1) Do they intersect? If so, then 0. If not,...
2) Which is greater? \[(x^{2}+1) - (x - x^{2}) > 0\]
3) If not, use \[(x-x^{2})-(x^{2}+1)\]. If so, How shall we find the minimum of the expression \[(x^{2}+1) - (x - x^{2})\]

Answer 2

The graphs do not intersect

Answer 3

I smell a first derivative coming.