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Calculus 3: Optimization

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Identifying Critical Points in a Multivariable Function

Given that f(x,y)=10−3x2−2y2+8y+12xf(x, y) = 10 - 3x^2 - 2y^2 + 8y + 12x, identify any critical points, saddle points, and any local extrema.

OptimizationMediumVideo
Critical Points of a Bivariate Quartic Function

Find and classify the critical points of f(x,y)=2x4+2y4−8xy+12f(x,y) = 2x^4 + 2y^4 - 8xy + 12.

OptimizationMediumVideo
Finding Absolute Extreme Values of a Function

Given the function f(x,y)f(x, y) on the rectangle D, find the absolute extreme values.

OptimizationMediumVideo
Wire Cut Optimization for Minimum and Maximum Area

A wire of length 100 centimeters is cut into two pieces; one is bent to form a square, and the other is bent to form an equilateral triangle. Where should the cut be made if (a) the sum of the two areas is to be a minimum; (b) a maximum? (Allow the possibility of no cut.)

OptimizationMediumVideo
Local Extrema Using Second Derivative Test

Find the local extrema of the function f(x,y)=x3−12xy+8y3f(x, y) = x^3 - 12xy + 8y^3 using the second derivative test.

OptimizationMediumVideo
Maximizing CobbDouglas Production Function with Constraint

Using the Cobb-Douglas production function 32L0.6K0.432L^{0.6}K^{0.4}, maximize production subject to the constraint 4L+2K=154L + 2K = 15.

OptimizationMediumVideo
Maximizing CobbDouglas Production Function with Constraint2

Using the Cobb-Douglas production function 32L0.6K0.432L^{0.6} K^{0.4}, maximize production subject to the constraint 4L+2K=154L + 2K = 15.

OptimizationMediumVideo
Optimizing TV Production Allocation with Cost Minimization

Suppose you make TV sets at two different factories, Factory A and Factory B. You produce x TVs at Factory A, y TVs at Factory B, and the cost given by the production of x TVs and y TVs is the function 6x2+12y26x^2 + 12y^2. You need to produce exactly 90 TV sets a month. Determine how many TVs should be produced at each factory to minimize the cost.

OptimizationMediumVideo