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Among all pairs of numbers (x,y) such that 3x+y=15, find the minimum of x^2 + y^2.

 Nov 17, 2020
 #1
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Solve  3x + y  =  15  for  y  to get  y = 15 - 3x

 

Let

 

z(x)  =  x2 + y2  =  x2 + (15 - 3x)2

 

So we want to find what value of  x  minimizes  z(x)

 

First let's find values of  x  which make  z'(x) = 0

 

z'(x)   =   2x + 2(15 - 3x)(-3)   =   2x - 90 + 18x   =   20x - 90  =  0  ⇒  x = 4.5

 

Now let's check whether the graph is concave up or down when  x = 4.5  using the second derivative test:

 

z''(x)  =  20   ⇒   z''(4.5)  =  20  >  0   so the graph is concave up

 

Since   z'(4.5)  =  0   and   z''(4.5) > 0  ,  we can say a min occurs when x = 4.5

 

When   x = 4.5,   y  =  15 - 3(4.5)  =  1.5

 

And   x2 + y2   =   (4.5)2 + (1.5)2   =   22.5

 

Check: https://www.desmos.com/calculator/fjdp793m19

 Nov 17, 2020

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