Let x,y>0 be real numbers and let s be the smallest of the numbers
x,y+(1/x),1/y
Find the greatest possible value of s .
For which x,y is this value assumed?
Again for beginners .. but experienced ones can answer if they want to..
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4 Answers
1x, y > 0 --> given
there can be four possible cases
1. x<1 y<1
here y+1/x>1 and 1/y>1 therefore s=x
2. x<1 y≥1
here s= x or 1/y depending on the values given to x and y
3. x≥1 y<1
same as above, s = x or 1/y
4. x≥1 y≥1
here s=1/y
among the four the s takes maximum value when s=1/y
the max value of s is 1
and y=1 and x ≥1
