Exercise 1600Calculus › Multivariable calculus › Optimisation in several variablesM1Added 3 Nov 2026

Constrained quadratic form

#quadratic form#Rayleigh quotient#eigenvalues

Statement

Maximize $f(x,y)=3x^2+2xy+3y^2$ on the unit circle.

Hint

The maximum is the largest eigenvalue of $\begin{pmatrix}3&1\\1&3\end{pmatrix}$ .

Show solutionHide solution

Eigenvalues. $(3-\lambda)^2-1=0\Rightarrow\lambda=2,4$ .

Maximum. $\lambda_{\max}=4$ , attained at the eigenvector $(1,1)/\sqrt2$ .

Conclusion. Maximum $4$ at $(\pm1/\sqrt2,\pm1/\sqrt2)$ with equal signs.