Chapter 6: Chapter 6

Find the maximum value of \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{xy}\); \((\mathrm{xy}>0)\) subject to the constraint \(\mathrm{x}^{2}+\mathrm{y}^{2}=8\) by drawing the level curves and by another method.

For the following quadratic form $$ \mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{ax}^{2}+2 \mathrm{bxy}+\mathrm{cy}^{2} $$ state conditions for \(\mathrm{f}(\mathrm{x}, \mathrm{y})\) to have a minimum and maximum value, using eigenvalues and the side condition \(\mathrm{x}^{2}+\mathrm{y}^{2}=1\).

Find the maximum and minimum of \(z=x^{2}+2 y^{2}-x\) on the set $x^{2}+y^{2} \leq 1$

Find the maximum and minimum of $\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{xy}-\mathrm{y}+\mathrm{x}-1$ in the closed disk \(\mathrm{D}=\\{\mathrm{P}:|\mathrm{P}| \leq 2\\}\) ( \(\mathrm{P}\) is a point in the \(\mathrm{xy}\) -plane).

Find the maximum value of the function $$ f(x, y)=4 x y-2 x^{3}-y^{4} $$ in the square $\mathrm{D}=\\{(\mathrm{x}, \mathrm{y}):|\mathrm{x}| \leq 2,|\mathrm{y}| \leq 2\\}$.

Find the points which might furnish relative maxima and minima of the function $$ \mathrm{f}(\mathrm{x}, \mathrm{y})=2 \mathrm{xy}-\left(1-\mathrm{x}^{2}-\mathrm{y}^{2}\right)^{3 / 2} $$ in the closed region \(\mathrm{x}^{2}+\mathrm{y}^{2} \leq 1\)

Find the extrema for the function \(\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}\) subject to the constraint \(x^{2}+2 y^{2}-z^{2}-1=0\)

Let the number 12 equal the sum of three parts $\mathrm{x}, \mathrm{y}, \mathrm{z}\(. Find values of \)\mathrm{x}, \mathrm{y}, \mathrm{z}$ so that \(\mathrm{xy}^{2} \mathrm{z}^{2}\) shall be a maximum (given the first condition and that \(\mathrm{x}, \mathrm{y}, \mathrm{z}>0)\).

Find the maximum of \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{xy}\) on the curve $$ G(x, y)=(x+1)^{2}+y^{2}=1 $$ assuming that such a maximum exists.

a) Find the maxima and minima of $\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+\mathrm{y}^{2}$ on the ellipse \(\mathrm{G}(\mathrm{x}, \mathrm{y})=2 \mathrm{x}^{2}+3 \mathrm{y}^{2}=1\) b) Find the maxima value of $\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{xyz}\( on the plane \)(x / a)+(v / b)+(z / c)=1(a, b, c>0)$