![[integral a to b of f(x)g(x)]^2 <= [integral a to b of f^2(x)] times [integral a to b of g^2(x)]](A6-eqn.gif)
if f(x) and g(x) are functions which are integrable on [a,b]. [6]
2. Consider the polynomial h(x) = x2 - x + 41. Note that h(1) = 41, h(2) = 43, h(3) = 47, h(4) = 53, and h(5) = 61 are all prime numbers. (An integer n > 1 is a prime number if its only positive integer divisors [i.e. factors] are 1 and n itself. For example 5 is prime, but 6 = 2 times 3 isn't.) Is h(x) always a prime number if x is a positive integer? Why or why not? [4]
Bonus. Is there any (other?) polynomial which always gives you prime numbers? [2]
Department of Mathematics
Trent University