$f\left(4\,0\right)\=13\/5$

$f\left(-2\,2\sqrt{3}\right)\=\left(13\+6\sqrt{3}\right)\/5\doteq 4.6785$

$f\left(-2\,-2\sqrt{3}\right)\=\left(13\+6\sqrt{3}\right)\/5\doteq 0.5215$

Table 4.10.4(a)   Boundary extrema

Initialize

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$f\left(x\,y\right)\=\left(x^2-xy\+y^2\+4y\+10\right)\/10$$\stackrel{\text{assign as function}}{\to }$$f$

Numerically find any unconstrained extrema

$f\left(x\,y\right)$

$\frac{1}{10}{x}^2-\frac{1}{10}xy+\frac{1}{10}{y}^2+\frac{2}{5}y+1$

$\stackrel{\text{minimize}}{\to }$

$\left[0.466666666666667\,\left[x=\mathrm{-1.33333333333333}\,y=\mathrm{-2.66666666666667}\right]\right]$

Analytically find any unconstrained extrema

$\frac{\partial }{\partial x} f\left(x\,y\right)\=0\,\frac{\partial }{\partial y} f\left(x\,y\right)\=0$

$\frac{x}{5}-\frac{y}{10}=0,-\frac{x}{10}+\frac{y}{5}+\frac{2}{5}=0$

$\stackrel{\text{solve}}{\to }$

$\left\{x=-\frac{4}{3}\,y=-\frac{8}{3}\right\}$

$f\left(-4\/3\,-8\/3\right)$ = $\frac{7}{15}$$$

Optimization Assistant: Optimize numerically subject to the boundary constraint

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$f\left(x\,y\right)\,x^2\+y^2\=16$

$\frac{1}{10}{x}^2-\frac{1}{10}xy+\frac{1}{10}{y}^2+\frac{2}{5}y+1,x^2+y^2=16$$$

Apply the boundary constraint analytically

$f\left(4 \cos \left(t\right)\,4 \sin \left(t\right)\right)$

$\frac{8{\cos \left(t\right)}^2}{5}-\frac{8\sin \left(t\right)\cos \left(t\right)}{5}+\frac{8{\sin \left(t\right)}^2}{5}+\frac{8\sin \left(t\right)}{5}+1$

$\stackrel{\text{simplify}}{\=}$

$\frac{13}{5}+\frac{\left(-8\cos \left(t\right)+8\right)\sin \left(t\right)}{5}$

$\stackrel{\text{differentiate w.r.t. t}}{\to }$

$\frac{8{\sin \left(t\right)}^2}{5}+\frac{\left(-8\cos \left(t\right)+8\right)\cos \left(t\right)}{5}$

$\stackrel{\text{roots}}{\to }$

$\left[0\,\frac{2\mathrm{\pi }}{3}\,\frac{4\mathrm{\pi }}{3}\,2\mathrm{\pi }\right]$

$f\left(4\,0\right)$$$ = $\frac{13}{5}$$\stackrel{\text{at 10 digits}}{\to }$$2.600000000$$$

$f\left(4 \cos \left(2 \mathrm{\π}\/3\right)\,4 \sin \left(2 \mathrm{\π}\/3\right)\right)$ = $\frac{13}{5}+\frac{6\sqrt{3}}{5}$$\stackrel{\text{at 10 digits}}{\to }$$4.678460970$ 

$f\left(4 \cos \left(4 \mathrm{\π}\/3\right)\,4 \sin \left(4 \mathrm{\π}\/3\right)\right)$ = $\frac{13}{5}-\frac{6\sqrt{3}}{5}$$\stackrel{\text{at 10 digits}}{\to }$$0.521539030$

Initialize

$f≔\left(x\,y\right)\to \left(x^2-xy\+y^2\+4y\+10\right)\/10\:$

Find any unconstrained extrema

$\mathrm{solve}\left(\left\{\mathrm{diff}\left(f\left(x\,y\right)\,x\right)\=0\,\mathrm{diff}\left(f\left(x\,y\right)\,y\right)\=0\right\}\right)$

$\left\{x=-\frac{4}{3}\,y=-\frac{8}{3}\right\}$

$f\left(-4\/3\,-8\/3\right)$ = $\frac{7}{15}$$$

Find extrema on the circumference of the disk

$\mathrm{Student}:-\mathrm{Calculus1}:-\mathrm{Roots}\left(\mathrm{diff}\left(f\left(4 \cos \left(t\right)\,4 \sin \left(t\right)\right)\,t\right)\=0\,t\=0..2 \mathrm{\π}\right)$

$\left[0\,\frac{2\mathrm{\pi }}{3}\,\frac{4\mathrm{\pi }}{3}\,2\mathrm{\pi }\right]$

$f\left(4\,0\right)$$$ = $\frac{13}{5}$$$

$f\left(4 \cos \left(2 \mathrm{\π}\/3\right)\,4 \sin \left(2 \mathrm{\π}\/3\right)\right)$ = $\frac{13}{5}+\frac{6\sqrt{3}}{5}$ 

$f\left(4 \cos \left(4 \mathrm{\π}\/3\right)\,4 \sin \left(4 \mathrm{\π}\/3\right)\right)$ = $\frac{13}{5}-\frac{6\sqrt{3}}{5}$$$