MA 110 - Solution to Assignment #6

Solution to Assignment #6

1. Prove that

[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)]

if f(x) and g(x) are functions which are integrable on [a,b].

[Solution to problem 1, part 1.]

[Solution to problem 1, part 2.]

[Solution to problem 1, part 3.]

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. Is h(x) always a prime number if x is a positive integer? Why or why not?

Solution. h(41) = 41² - 41 + 41 = 41².

Bonus. Is there any (other?) polynomial which always gives you prime numbers?

[Solution to bonus problem.]

Assignment #6

$Math logo$ Department of Mathematics

Trent University

Maintained by Stefan Bilaniuk. Last updated 1998.08.22.