Question

Which best describes the graph of the cubic function f(x) = x3 + x2 + x + 1?

As x increases, y increases along the entire graph.
As x increases, y increases, decreases, and then increases again.
As x increases, y decreases, increases, and then decreases again.
As x increases, y decreases along the entire graph.

Answer 1

As x increases, y increases along the entire graph.

Reason: f'(x) = 3x^2 + 2x + 1. Its discriminant is 2^2 − 4·3·1 = −8 < 0, so f'(x) > 0 for every real x. Thus f is strictly increasing everywhere.