Exercise 857Calculus › Multivariable calculus › Optimisation in several variablesL1Added 3 Nov 2026

Boundary classification

#boundary#classification#circle

Statement

Find and classify the extrema of $f(x,y)=xy$ on the unit circle $x^2+y^2=1$ .

Hint

Parametrize the circle; $f=\frac12\sin 2\theta$ .

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Parametrization. $x=\cos t$ , $y=\sin t$ , $f=\cos t\sin t=\frac12\sin 2t$ .

Extrema. $f$ ranges in $[-1/2,1/2]$ .

Points. Max $1/2$ at $t=\pi/4, 5\pi/4$ ; min $-1/2$ at $t=3\pi/4, 7\pi/4$ .

Conclusion. Maximum $1/2$ at $(\pm1/\sqrt2,\pm1/\sqrt2)$ with matching signs; minimum $-1/2$ with opposite signs.