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Maximize Values - Calculus










































Find two non negative numbers x and y such that 2x+2y=500 and x^3+y^3 is maximal.

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Apr 09, 2012
Maximize Values - Calculus
by: Staff


Question:

Find two non negative numbers x and y such that 2x+2y=500 and x^3+y^3 is maximal.


Answer:


Maximize the function f(x,y) = x³ + y³


subject to the following constraint:

2x + 2y = 500

Since 2x + 2y = 500 is a linear equation, the only possible non-negative values of x and y can be are:

0 ≤ y ≤ 250

0 ≤ x ≤ 250


2x + 2y = 500

Solving for y

y = 250 - x


substitute 250 – x for the y value in f(x,y)

f(x,y) = x³ + y³

f(x) = x³ + (250 – x)³

f(x) = 750x² - 187500x + 15625000

f(x) = 750(x – 125)² + 3906250


as you can see, once the function is expanded, it turns out to be a parabola




f(x,y) will be at its minimum of 3906250 (the vertex) when x = 125 and y = 125


The question is: What is the maximum given the following constraints:

0 ≤ y ≤ 250

0 ≤ x ≤ 250



>>> the final answer is:

There are two solutions:

Given the constraints on the values of x and y, f(x,y) = x³ + y³ is maximized when:

x = 0 and y = 250

or

x = 250 and y = 0



Thanks for writing.

Staff
www.solving-math-problems.com


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