Mathematics 7356: Proof Topical Past Paper Questions

Prove that the graph of the curve with equation $$ y=x^{3}+15x-\frac{18}{x} $$ has no stationary points.

Prove that the sum of the cubes of two consecutive odd numbers is always a multiple of 4. [5 marks]

Integers m and n are both odd. Prove that $ m^2 + n^2 $ is a multiple of 2 but not a multiple of 4 [5 marks]