Find the minimum value of (x+1x)6−(x6+1x6)−2(x+1x)3+(x3+1x3) for x > 0.
The expression has a minimum value of 6, when x = 1.
(x+1x)6−(x6+1x6)−2(x+1x)3+(x3+1x3)
Just by inspection I can see that as x moves away from 1 in either direction the function value will increase.
So the minimum will occur when x=1
that minimum will be
26−2−223+2=6010=6
Just as our guest said,