SOLUTION: Among all pairs of numbers (x,y) such that 4x+y=18, find the pair for which the sum of squares, x2+y2, is minimum. Write your answers as fractions reduced to lowest terms.

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Question 1125614: Among all pairs of numbers (x,y) such that 4x+y=18, find the pair for which the sum of squares, x2+y2, is minimum. Write your answers as fractions reduced to lowest terms.
Found 2 solutions by jim_thompson5910, ikleyn:
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!

Solve for y

Subtract 4x from both sides

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Now plug this into the function

Notice how y is replaced with -4x+18

Expand

FOIL

Combine like terms

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The goal is to minimize z, so we want to find the smallest z value for some input x

The equation is in the form with a = 17, b = -144, c = 324

The x coordinate of the vertex is . Let's plug in the values for a and b.

This is the x value that makes z the smallest. So which leads to

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So the final answer is that and . This pair of values makes the smallest possible, ie a minimum.

Answer by ikleyn(52803) (Show Source):
You can put this solution on YOUR website!
.

They want you find the minimum of this quadratic function f(x) = + = = . For the general form of a quadratic function y = ax^2 + bx + c the minimum is achieved at x = . In your case the minimum is at x= = . The pair under the question is x = , y = = .

Solved.