Mathematico | Mathematico

You are given that

x + y = 6

Find the minimum possible value of

2 x^{2} + y^{2}

Hint

x + y = 6 \Rightarrow y = 6 - x

Hint

Substitute $y = 6 - x$ into $2 x^{2} + y^{2},$ then complete the square to find the minimum value.

Solution

We have

x + y = 6 \Rightarrow y = 6 - x

\begin{aligned} 2 x^{2} + y^{2} & = 2 x^{2} + (6 - x)^{2} \\ = 2 x^{2} + x^{2} - 12 x + 36 \\ = 3 x^{2} - 12 x + 36 \\ = 3 [x^{2} - 4 x] + 36 \\ = 3 [(x - 2)^{2} - 4] + 36 \\ = 3 (x - 2)^{2} + 24 \end{aligned}

We see that the minimum value is $24$ and occurs when $x = 2$ and $y = 4$ .