Is there a maxima or minima for a parabola opening left, right or diagonal?

Question

If a parabola is opening up or down, we can use the vertex to determine if it has maxima or minima. How do we identify the same in case of a parabola opening left, right or diagonal?

Answer 1

Interesting question. The short answer is that a parabola pointed in any direction has either a maximum or a minimum and that these are finite except for a parabola pointed straight up/down.

To see this, change the problem so that instead of rotating the parabola, we're rotating the the direction. That is, consider the basic parabola

$$y = x^2$$

and ask the furthest distance it travels in an arbitrary direction

$$(\cos \theta, \sin \theta)$$.

It will reach this furthest distance when this direction is normal to the parabola, which is also when the perpendicular direction,

$$(-\sin \theta, \cos \theta)$$

is tangent to our parabola.

However, by using calculus or drawing a picture, we can see that tangent lines of any slope except purely vertical contact the parabola at some point. We can also show that the parabola, since it has positive curvature, lies strictly above the tangent line. Thus, the original parabola has a maximum extent in any direction except the $y$-direction and therefore a parabola at any other orientation should have either a maximum or a minimum $y$-value.