Paper 2, Section I, 2C2 \mathrm{C}

Differential Equations | Part IA, 2017

Consider the function

f(x,y)=xy+yx(xy)2a2f(x, y)=\frac{x}{y}+\frac{y}{x}-\frac{(x-y)^{2}}{a^{2}}

defined for x>0x>0 and y>0y>0, where aa is a non-zero real constant. Show that (λ,λ)(\lambda, \lambda) is a stationary point of ff for each λ>0\lambda>0. Compute the Hessian and its eigenvalues at (λ,λ)(\lambda, \lambda).

Typos? Please submit corrections to this page on GitHub.