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Magazine Problem 1
Use Lagrange multipliers to find the maximum and minimum values of \(f\) subject to the given constraint, if such values exist. $$f(x, y)=x+y, \quad x^{2}+y^{2}=1$$
Problem 2
Use Lagrange multipliers to find the maximum and minimum values of \(f\) subject to the given constraint, if such values exist. $$f(x, y)=x+3 y+2, \quad x^{2}+y^{2}=10$$
Problem 3
Use Lagrange multipliers to find the maximum and minimum values of \(f\) subject to the given constraint, if such values exist. $$f(x, y)=(x-1)^{2}+(y+2)^{2}, \quad x^{2}+y^{2}=5$$
Problem 4
Use Lagrange multipliers to find the maximum and minimum values of \(f\) subject to the given constraint, if such values exist. $$f(x, y)=x^{3}+y, \quad 3 x^{2}+y^{2}=4$$
Problem 4
A function \(f(x, y)\) has partial derivatives \(f_{x}(1,2)=3\) \(f_{y}(1,2)=5 .\) Explain how you know that \(f\) does not have a minimum at (1,2).
Problem 5
Use Lagrange multipliers to find the maximum and minimum values of \(f\) subject to the given constraint, if such values exist. $$f(x, y)=3 x-2 y, \quad x^{2}+2 y^{2}=44$$
Problem 5
Without calculus, find the highest and lowest points (if they exist) on the surface. The z-axis is upward. $$x^{2}+y^{2}+(z-1)^{2}=49$$
Problem 6
Without calculus, find the highest and lowest points (if they exist) on the surface. The z-axis is upward. $$(x+1)^{2}+(y-3)^{2}+2 z^{2}=162$$
Problem 6
Use Lagrange multipliers to find the maximum and minimum values of \(f\) subject to the given constraint, if such values exist. $$f(x, y)=x y, \quad 4 x^{2}+y^{2}=8$$
Problem 6
The function has a critical point at (0,0) What sort of critical point is it? $$f(x, y)=x^{2}-\cos y$$
