THE POSITIVE NUMBERS x and y SATISFY xy=1.THE MINIMUM VALUE OF
1/x^4 +1/4y^4.
I think this is a direct problem of app. of derivative(maxima & minima)
Just substitute y=1/x in 2nd statement & find its minimum value using 2nd derivative approach.
The answer I am getting is 1.
Use A.M.-G.M. approach.Since both are positive numbers, a.m.-g.m. inequality is going to work here.[1/x4 +1/y4]/2> or= {1/x4y4}1/2 and xy=1 so1/x4 +1/y4 > or=2 .So, the answer is 2.
In the statement we have to find min. value of 1/x4+1/4y4 not of 1/x4 +1/y4 ,but your method is correct ,by this method also answer comes out is 1.