Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Trending News
analysis question, please help?
let A=[0,∞)⊆R.
prove that for all n∈N the function x→x^(1/n):A→R is increasing and continuous
I really need a help with this question, thank you so much!
2 Answers
- ?Lv 68 years agoFavorite Answer
we have y=x^(1/n)
since both sides are positive we can take logs
ln(y)=(1/n)ln(x). differentiate:
y ' / y = 1 / (nx)
y ' = y / (nx) = x^(1/n) / (nx), which is positive, since n and x are positive, so y is increasing
- ?Lv 78 years ago
The function f:AâR defined by f(x) = x^(1/n)
Since f(x) = exp((1/n) ln(x)), ln is continuous on A, and exp is continuour on R, then x^(1/n) is continuous on A.
f '(x) = (1/n) x^(-(1 - 1/n)) > 0 for all x in A.
So f is increasing on A.
