 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/12/c/ii/solution

Now

h = e^{x} - e^{y} - x + y

and the origin has stationary value zero. Near it,

h = \frac{x ^{2} - y ^{2}}{2} + \frac{x ^{3} - y ^{3}}{6} + O ((∣ x ∣ + ∣ y ∣)^{4}) .

x

-axis

h = e^{x} - x - 1 > 0

for

x \neq = 0

, while along the

y

-axis

h = 1 - e^{y} + y < 0

for

y \neq = 0

. The zero level set includes the exact diagonal

x = y

and another branch crossing it at the origin; this second branch is tangent to

y = - x

but is curved away from the origin. The right panel of the contour figure displays these features.

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