91Ó°ÊÓ

Find the extreme values of \(f(x, y)=x y\) subject to the constraint \(g(x, y)=x^{2}+y^{2}-10=0\)

Find the points on the curve \(x^{2}+x y+\) \(y^{2}=1\) in the \(x y\) -plane that are nearest to and farthest from the origin.

Find the radius and height of the open right circular cylinder of largest surface area that can be inscribed in a sphere of radius \(a .\) What is the largest surface area?

Use the method of Lagrange multipliers to find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse \(x^{2} / 16+y^{2} / 9=1\) with sides parallel to the coordinate axes.

(a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine if the domain is an open region, a closed region, or neither, and ( \(\mathbf{f}\) ) decide if the domain is bounded or unbounded. $$f(x, y)=4 x^{2}+9 y^{2}$$

Find all the local maxima, local minima, and saddle points of the functions. $$f(x, y)=e^{y}-y e^{x}$$

Find the volume of the largest closed rectangular box in the first octant having three faces in the coordinate planes and a vertex on the plane \(x / a+y / b+z / c=1\) where \(a>0, b>0,\) and \(c>0.\)

Find the absolute maxima and minima of the functions on the given domains. \(f(x, y)=2 x^{2}-4 x+y^{2}-4 y+1\) on the closed triangular plate bounded by the lines \(x=0, y=2, y=2 x\) in the first quadrant.

In economics, the usefulness or utility of amounts \(x\) and \(y\) of two capital goods \(G_{1}\) and \(G_{2}\) is sometimes measured by a function \(U(x, y) .\) For example, \(G_{1}\) and \(G_{2}\) might be two chemicals a pharmaceutical company needs to have on hand and \(U(x, y)\) the gain from manufacturing a product whose synthesis requires different amounts of the chemicals depending on the process used. If \(G_{1}\) costs \(a\) dollars per kilogram, \(G_{2}\) costs \(b\) dollars per kilogram, and the total amount allocated for the purchase of \(G_{1}\) and \(G_{2}\) together is \(c\) dollars, then the company's managers want to maximize \(U(x, y)\) given that \(a x+b y=c .\) Thus, they need to solve a typical Lagrange multiplier problem. Suppose that $$U(x, y)=x y+2 x$$ and that the equation \(a x+b y=c\) simplifies to $$2 x+y=30$$ Find the maximum value of \(U\) and the corresponding values of \(x\) and \(y\) subject to this latter constraint.

Minimize the function \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) subject to the constraints \(x+2 y+3 z=6\) and \(x+3 y+9 z=9.\)