Test site: basic inequality.

Special topic: computational problems; Change your mind.

Analysis: Let t=2x+y, express the known equation by t, and arrange it into a quadratic equation about X. The quadratic equation has a solution, and the discriminant is greater than or equal to 0. Find the range of t and the maximum value of 2x+y. 。

Solution: Solution: ∫4x 2+y2+xy = 1.

∴(2x+y)2-3xy= 1

Let t=2x+y and y=t-2x.

∴t2-3(t-2x)x= 1

That is 6x2-3tx+t2- 1=0.

∴△=9t2-24(t2- 1)=- 15t2+24≥0

solve

What is the maximum value of 2x+y?

So the answer is

Comments: This question examines that quadratic equation has method of substitution solution, and the number of quadratic equation solutions is determined by discriminant.