Quiz 7, Complete Solution for Problem 3

We are looking for the points where the gradient of f is parallel to the gradient of g, or in other words the points where the gradient of f is orthogonal to the curve. (Why?)

In plot below we have marked with dots the two points where the gradient of f is orthogonal to the curve:

It's fairly clear that the gradient is orthogonal to the curve at the upper point. At the lower point our answer represents a best guess, as the arrows tilt a little to the right on the right of the point, and a little to the left to the left of the point.

Which point is the local max, and which is the local min?

We can use the direction of the gradient field to find out.

At the lower point, imagine how the levels of f look, by imagining curves perpendicular to the gradient of f. If you are at the lower point, and move in the direction of the gradient, you can see that the value of f must increase, because higher levels of f are hitting the curve. So, f has a local min at the lower point.

At the upper point, if you move in the direction of the grdient, the higher levels of f no longer touch the curve, so the value of f has reached its max at the upper point.

Return to Quiz 7.