Question

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If x and y be two real variables such that x>0 and xy=1. Then, find the minimum value of x+y.

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Answer 1

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Answer 2

Solution:-As we know that, according to the concept of Maxima and minima of diffrentiation..

if f(x) is any equation......

then,if

f'(x)=0

and ,f''(x)>0,then that value of x will give maximum value but if we get f''(x)<0 ,then the value of x will give minimum value

now here

xy=1

y=1/x

so, putting this value in x+y and taking it as f(x) we can write

f(x)=x+1/x=x+x^-1

diffrentiating with respect to x we get

f'(x)=1+(-1)x^-2

f'(x)=1-1/x^2

clearly we can say that,f(1)=0

f(1)=1-1=0

diffrentiating it again

f''(x)=0-(-2)/x^3=2/x^3

f''(1)=2/1=2

so,from here we are getting

f'(1)=0 and f''(1)>0

so,x=1 will give minimum value

x+y=x+1/x=1+1/1=2=minimum value