Thursday, February 21 ∗∗ Constrained min/max via Lagrange multipliers.1. Let C be the curve in R2given by x3+ y3= 16.(a) Sketch the curve C.(b) Is C bounded?(c) Is C closed?2. Consider the function f (x, y) = ex yon C.(a) Is f continuous? What does the Extreme Value Theorem tell you about the existance ofglobal min and max of f on C?(b) Use Lagrange multipliers to determine both the min and max values of f on C .3. Consider the surface S given by z2= x2+ y2(a) Sketch S.(b) Use Lagrange multipliers to find the points on S that are closest to (4,2,0).4. For the function shown on the back of the sheet, use the level curves to find the locations andtypes (min/max/saddle) for all the critical points of the function:f (x, y) = 3x − x3− 2y2+ y4Use the formula for f and the 2nd-derivative test to check your answer.5. If the length of the diagonal of a rectangular box must be L, what is the largest possible
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