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Let xa n dy be two real variable such th...

Let `xa n dy` be two real variable such that `x >0a n dx y=1` . Find the minimum value of `x+y` .

Text Solution

Verified by Experts

The correct Answer is:
2
Let ` f (x)= x + y`, where ` xy = 1`
` rArr " " f (x) = x + (1)/(x)`
` rArr " "f ' (x) = 1 - (1)/(x^(2)) = (x ^(2) - 1)/( x ^(2))`
Also, ` " f '' (x) = 2 //x ^(3)`
On putting ` f' (x) = 0`, we get
`" " x = pm 1, " but " x gt 0 `[ neglecting ` x = -1 `]
` f '' (x) gt 0, " for " x = 1 `
Hence, ` f (x)` attains minimum at ` x = 1, y = 1 `
`rArr (x + y )` has minimum value 2.
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