Determine all relative extrema for the following function fx

Determine all relative extrema for the following function f(x)=2x-(256/x)

Solution

At a relative extrema, a function changes from increasing to decreasing or decreasing to increasing. In order to determine the critical points of the function f(x) = 1x – 256/x, let us set f (x) = 0. Then, 2x – 256/x = 0 or, 2x² – 256 = 0 or, 2x² = 256 or, x² = 128 so that x = ± 82. Also, f(x) is defined at both the critical points x = ± 82. Further, f’ = 2 – 256( -1)x^-2 = 2+256/x² . Since f’ does not change sign at any of the critical points x = ± 82 , there is no relative extrema at these critical points. Thus the function f (x) has no relative extrema.