Suppose n is some positive integer, and f is the function f(x) = nx^(x^2). Say f
ID: 3342703 • Letter: S
Question
Suppose n is some positive integer, and f is the function f(x) = nx^(x^2). Say for example, when n = 5, f(x) = 5x^25 and when n = 7, f(x) = 7x^49. Using this knowledge:
(i) What is the SMALLEST value of f on the unit interval, [0,1]? Your answer won't depend on n.
(ii) What is the LARGEST value of f on the unit interval, [0,1]? Your answer will depend on n. What happens to this value as n goes to infinity?
(iii) What is the AVERAGE value of f on the unit interval, [0,1]? Your answer will depend on n. What happens to this value as n goes to infinity?
(iv) Your answer to (b) and (c) show that the largest value of f and the average value of f are different when n is large. Briefly explain how this is possible.
Explanation / Answer
since f'(x) >0 for all [0,1] so f(x) keeps increasing..so min is at 0 and max at 1
a)0
1)n.It goes to infinity as n goes to infinity
c)avg={integral 0 to1 n x^(n^2)}/(1-0)=n/(n^2+1).It goes to zero as n tends to infinity
d)when is large(towards infinity) we see largest value and average are different.n is not equal to n/(n^2+1) for large n.This is because f increases very slowly in [0,1] as n tends to become larger..so avg is close to zero .