Calculus or Algebra?

Algebra Level 3

f ( x , y ) = x 2 + 13 y 2 − 6 x y − 4 y − 2 f(x,y) = x^{2} + 13y^{2} - 6xy - 4y - 2

Let x x and y y be real numbers . If the minimum value of f ( x , y ) f(x,y) occurs when ( x , y ) = ( A , B ) (x,y) = (A,B) , find A + B A+B .


The answer is 2.

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Rishabh Tiwari by Rishabh Tiwari , 20, India

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1 solution

Completing the square : \text{Completing the square :}

f ( x , y ) = x 2 + 13 y 2 − 6 x y − 4 y − 2 = ( x − 3 y ) 2 + ( 2 y − 1 ) 2 − 3 \displaystyle f(x,y)=x^2+13y^2-6xy-4y-2 = (x-3y)^2 + (2y-1)^2 - 3

So , f ( x , y ) ≥ − 3 \displaystyle f(x,y) \ge -3 Since ( x − 3 y ) 2 ≥ 0 , ( 2 y − 1 ) 2 ≥ 0 \displaystyle (x-3y)^2 \ge 0 , (2y-1)^2\ge 0

So minimum occurs when { x = 3 y = 3 2 y = 1 2 \displaystyle \begin{cases} x=3y=\frac{3}{2} \\ y=\frac{1}{2} \end{cases}

Thus the answer 3 2 + 1 2 = 2 \boxed{\frac{3}{2}+\frac{1}{2}=2}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Typo: The answer is 2 2

Hung Woei Neoh - 5 years ago

Yup. Nice solution! +1 , Thanks for solving it!

Rishabh Tiwari - 5 years ago

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Yes welcome. It has solutions using calculus and that's more beautiful.

Aditya Narayan Sharma - 5 years ago

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Yes , I would like to see a good solution(like yours) with calculus in it , @Aditya Sharma !

Rishabh Tiwari - 5 years ago

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