Exercise 1598Calculus › Multivariable calculus › Optimisation in several variablesL1Added 3 Nov 2026

Global extrema on a compact set

#compactness#global extrema#boundary

Statement

Find the global extrema of $f(x,y)=x^2-y^2$ on the disk $x^2+y^2\le1$ .

Hint

Check the interior (Hessian indefinite), then the boundary (polar).

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Interior. $\nabla f=0\Rightarrow(0,0)$ , saddle (Hessian indefinite).

Boundary. With $x=\cos t,y=\sin t$ : $f=\cos^2t-\sin^2t=\cos 2t$ .

Values. Range $[-1,1]$ , with max $1$ at $(1,0)$ and min $-1$ at $(0,1)$ .

Conclusion. Global max $1$ , global min $-1$ on the boundary.