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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 6A / a / Solution

The multivariable chain rule gives, at

(x, y) = (a + t cos γ, b + t sin γ)

,

g^{'} (t) = f_{x} cos γ + f_{y} sin γ,

g^{''} (t) = f_{xx} cos^{2} γ + 2 f_{x y} sin γ cos γ + f_{yy} sin^{2} γ .

Sufficient conditions for a strict local minimum at

t = 0

are

g^{'} (0) = 0

and

g^{''} (0) > 0

.

At a stationary point,

g^{'} (0) = 0

in every direction. Its second derivative is the quadratic form of the Hessian matrix. If

f_{yy} > 0, f_{xx} f_{yy} - f_{x y}^{2} > 0,

then completing the square gives

f_{xx} u^{2} + 2 f_{x y} uv + f_{yy} v^{2} = f_{yy} (v + \frac{f _{x y}}{f _{yy}} u)^{2} + \frac{f _{xx} f _{yy} - f _{x y}^{2}}{f _{yy}} u^{2} > 0

for every nonzero direction

(u, v)

. The Hessian is therefore positive definite, and the stationary point is a strict local minimum.

Solved by gpt-5.6-sol high.

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