 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/12/d/solution

The zero level set is especially simple:

f = 0 ⟺ x = 0 or y = 0 or x^{2} + y^{2} = 1 .

The two axes and the unit circle divide the plane into regions where the signs alternate. Inside the circle,

f

has the opposite sign to

x y

; outside it has the same sign. Closed level curves surround the two interior local minima and two interior local maxima; the zero curves cross at the five saddle points of a scalar function. Farther out, the nonzero level curves are open within the displayed square.

 New to topics? Read the docs here!