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What is the minimum value of the expression 2x^2 + 3y^2 + 8x - 10y + 62 for real x and y?

 May 29, 2021
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2x^2 + 3y^2 + 8x - 10y + 62

(2x^2 + 8x) + 3y^2 - 10y + 62

(2(x+2)^2 - 8) + (3(y - 5/3)^2 - 25/3) + 62

2(x+2)^2 + 3(y - 5/3)^2 + 45 2/3

Quadratics have to be non-negative, meaning that 2(x+2)^2 and 3(y - 5/3)^2 have the lowest value of 0. 

0 + 0 + 45 2/3 = 45 2/3. 

 

=^._.^=

 May 29, 2021

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