\(\displaystyle{\left({2}{x}+\sqrt{{{4}{x}^{{2}}+{1}}}\right)}{\left(\sqrt{{{y}^{{2}}+{4}}}-{2}\right)}\ge{y}{>}{0}\) minimum vale of \(\displaystyle{x}+{y}\)

Polchinio3es

Polchinio3es

Answered question

2022-03-30

(2x+4x2+1)(y2+42)y>0 minimum vale of x+y

Answer & Explanation

crazyrocketrz5z

crazyrocketrz5z

Beginner2022-03-31Added 11 answers

Explanation:
g(y)=yy2+42 is monotonic decreasing and g(y)1 for y0, and function f(x)=2x+4x2+1 is monotonic increasing and f(x)1 requires x0.
First we could easily find out that when x+y is minimized,
f(x)=g(y)=z1 so x=f1(z)=z214z and y=g1(z)=4zz21.
So x+y=z214z+4zz212, where z1
The minimal value is reached when z21=4z.

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