Task

Algebra Level 4

f ( x , y ) = 14 x 2 + 9 y 2 + 22 x y − 42 x − 34 y + 35 \large f(x,y)=14x^2+9y^2+22xy-42x-34y+35

Let reals x x and y y satisfying x ≤ 2 x\leq 2 and x + y ≥ 2 x+y \geq 2 .

Find the minimum value of f ( x , y ) f(x,y) .

Give your answer to 2 decimal places.

This problem was taken from Bac Ninh grade 10 entrance test.


The answer is 3.00.

P C by P C , 20, Vietnam

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1 solution

Chew-Seong Cheong
Jun 10, 2016

f ( x , y ) = 14 x 2 + 9 y 2 + 22 x y − 42 x − 34 y + 35 x + y ≥ 2 ⟹ y ≥ 2 − x ≥ 14 x 2 + 9 ( 2 − x ) 2 + 22 x ( 2 − x ) − 42 x − 34 ( 2 − x ) + 35 ≥ 14 x 2 + 9 x 2 − 36 x + 36 − 22 x 2 + 44 x − 42 x + 34 x − 68 + 35 ≥ x 2 + 3 Since x 2 ≥ 0 ≥ 3 \begin{aligned} f(x,y) & = 14x^2+9\color{#3D99F6}{y}^2+22x\color{#3D99F6}{y}-42x-34\color{#3D99F6}{y}+35 \quad \quad \small \color{#3D99F6}{x+y \ge 2 \implies y \ge 2-x} \\ & \color{#3D99F6}{\ge}14x^2+9\color{#3D99F6}{(2-x)}^2+22x\color{#3D99F6}{(2-x)}-42x-34\color{#3D99F6}{(2-x)}+35 \\ & \ge 14x^2+ 9x^2-36x+36 - 22x^2+44x - 42x + 34x - 68 + 35 \\ & \ge \color{#3D99F6}{x^2} + 3 \quad \quad \small \color{#3D99F6}{\text{Since }x^2 \ge 0} \\ & \color{#3D99F6}{\ge} \boxed{3} \end{aligned}

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