Let a>b>0 and let n be in N (natural numbers) satisfy n greater than or equal to 2. Prove that:
a^(1/n)-b^(1/n) < (a-b)^(1/n) by showing that x^(1/n)-(x-1)^(1/n) is decreasing for all x greater than or equal to 1 and using that property appropriately.
Thank you for any help.
December 2nd 2006, 07:21 PM
Taking the derivative of what they gave x^(1/n)-(x-1)^(1/n),
(1/n)[(x^((1-n)/n))-(x-1)^((1-n)/n)). How do I show that this is decreasing?
I'm assuming that then I can say that using this,
b=x-1 (since b is less than a)
and use the derivatives accordingly. Also how can I conclude that the right side is less than the left in the original equation I want to prove? Is it just because it's decreasing?
December 2nd 2006, 07:27 PM
Originally Posted by dave007rules
How do I show that this is decreasing?
Since the function is differenciable you show that the derivative is zero.