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Magazine Problem 36
Find the indicated maximum or minimum values of \(f\) subject to the given constraint. Maximum: \(f(x, y, z)=x+2 y-2 z ; \quad x^{2}+y^{2}+z^{2}=4\)
Problem 37
Find the indicated maximum or minimum values of \(f\) subject to the given constraint. Maximum: \(f(x, y, z, t)=x+y+z+t\) \(x^{2}+y^{2}+z^{2}+t^{2}=1\)
Problem 37
Use a 3 D graphics program to generate the graph of each function. $$f(x, y)=x^{3}-3 x y^{2}$$
Problem 37
Find \(f_{x x}, f_{x y}, f_{y x},\) and \(f_{y y}\) (Remember, \(f_{y x}\) means to differentiate with respect to \(y\) and then with respect to \(x\).) $$f(x, y)=x+e^{y}$$
Problem 38
Find \(f_{x x}, f_{x y}, f_{y x},\) and \(f_{y y}\) (Remember, \(f_{y x}\) means to differentiate with respect to \(y\) and then with respect to \(x\).) $$f(x, y)=y-e^{x}$$
Problem 38
Find the indicated maximum or minimum values of \(f\) subject to the given constraint. Minimum: \(f(x, y, z)=x^{2}+y^{2}+z^{2} ; x-2 y+5 z=1\)
Problem 38
Use a 3 D graphics program to generate the graph of each function. $$f(x, y)=\frac{1}{x^{2}+4 y^{2}}$$
Problem 39
Find \(f_{x x}, f_{x y}, f_{y x},\) and \(f_{y y}\) (Remember, \(f_{y x}\) means to differentiate with respect to \(y\) and then with respect to \(x\).) $$f(x, y)=y \ln x$$
Problem 40
Business: maximizing production. A computer company has the following Cobb- Douglas production function for a certain product: $$ p(x, y)=800 x^{3 / 4} y^{1 / 4} $$ where \(x\) is the labor, measured in dollars, and \(y\) is the capital, measured in dollars. Suppose that the company can make a total investment in labor and capital of \(\$ 1,000,000\) How should it allocate the investment between labor and capital in order to maximize production?
Problem 40
Find \(f_{x x}, f_{x y}, f_{y x},\) and \(f_{y y}\) (Remember, \(f_{y x}\) means to differentiate with respect to \(y\) and then with respect to \(x\).) $$f(x, y)=x \ln y$$
