x> 0, Y > 0, X / 2 + Y / 5 = 2, lgx + lgY max

x> 0, Y > 0, X / 2 + Y / 5 = 2, lgx + lgY max

To get the meaning of a question
2=x/2+y/5≥2√(x/2*y/5)
∴x/2*y/5≤1
∴xy≤10
∴lgx+lgy=lg(xy)≤lg10=1
The maximum value of lgx + lgY is 1

Let x > 0, Y > 0, and 2x + y = 20, then the maximum value of lgx + lgY is______ ．

∵ x > 0, Y > 0, and 2x + y = 20 ∵ 2x + y = 20 ≥ 22xy, (if and only if 2x = y, the equal sign holds) ∵ XY ≤ 50lgx + lgY = LG (XY) ≤ LG50 = 1 + lg5. That is, the maximum value of lgx + lgY is 1 + lg5. So the answer is 1 + lg5

If (lgx) + (lgY) = lgx + lgY, then the value range of XY is given

If x > = 1, Y > = 1, and XY > = 10, x ^ (lgx) * y ^ (lgY) > = 10, then the value of X + y is
Let me first give my approach:
Taking logarithm on both sides, we get (lgx) ^ 2 + (lgY) ^ 2 > = 1
And xy = 10, logx + lgY = 1
From the above two formulas, we get lgx * lgY = 1, Y > = 1, so lgx > = 0, lgY > = 0
Here, the problem is almost solved, but I'm stuck here=|||
I see some solutions that say x = 1, y = 10 or x = 10, y = 1, in short, x + y = 11
Why? As long as one of X and Y is equal to 1, it is consistent with [3]. Why does the other have to be equal to 10? It doesn't matter how much it is?
One wrong word, eh

What do you do next
x>=1,y>=1
Then lgx ≥ 0, lgY ≥ 0, so lgx * lgY ≥ 0
And lgx * lgY