Texts: Please do not say "solve for this to get the answer" like the other answers to this question. Please solve for the answers and explain. This problem will guide you through the optimization of f(x, y) = x^3 + y^3 + 6xy - 8y over the rectangle R = [-4, 4] × [-3, 3].
This problem will guide you through the optimization of f(x, y) = x^3 + y^3 + 6xy - 8y over the rectangle R = [4, 4] × [3, 3].
This problem will have you type in lots of lists. A list of numbers should be separated with commas: 3, -4, √7 for example. A list of points should appear as Cartesian coordinates and be separated with commas: (3, 2), (-4, 7), (√2, √7) for example. If a list is empty, type NONE.
The next few steps will have you find boundary critical points.
(b) Critical points on the bottom
Answer from 4 to 4 f(x, 3) =
Differentiate this (-3x^2 + 6y) = 0 when x = [] Only list solutions with x in [4, 4] like (3, 2), (4, 5).
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Differentiate the equation to get the critical points within [4, 4]. As we want to minimize the function: ∂f/∂y = 0.
Differentiate this 4y - 8 = 0 when y = [] Only list solutions with y in [3, 3]. As a result,
(e) Corner points The maximum
List all of the candidates to optimize and evaluate. List all of the candidates to optimize. Evaluate f at all of these points.
Make a conclusion The function f(x, y) = x^3 + y^3 + 6xy - 8y has a maximum value over R = [4, 4] × [3, 3] or is achieved at the point (x, y) = [].
The function f(x, y) = x^3 + y^3 + 6xy - 8y has a minimum value over R = [4, 4] × [3, 3] and is achieved at the point (x, y) = [].
