 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/a/solution

a e^{x} = c

and

b e^{y} = d

. The sign assumptions make both ratios positive, so each equation has exactly one real solution. Therefore

(x_{*}, y_{*}) = (lo g \frac{c}{a}, lo g \frac{d}{b}) .

Substitution gives the stationary value

h_{*} = c + d - c lo g \frac{c}{a} - d lo g \frac{d}{b} .

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