Question

Let f(x)=2x+x2 (A) Use interval notation to indicate where f(x) is increasing. Note: Use 'INF' for [infinity], '-INF' for −[infinity], and use 'U' for the union symbol. Increasing: (B) Use interval notation to indicate where f(x) is decreasing

Answer 1

The function f(x) = 2x + x^2 is increasing on the interval (0, INF) and would be decreasing on the interval (-INF, -1) if we consider x < 0.

Step-by-step explanation:

To determine where the function f(x) = 2x + x^2 is increasing or decreasing, we first need to find its derivative, which gives us the slope of the function at any given point. The derivative of f(x) is f'(x) = 2 + 2x. This derivative is always positive, since 2 is positive and 2x is positive for all x > 0. Therefore, f(x) is increasing over the interval (0, INF). As there are no points where f'(x) is negative, there are no intervals where f(x) is decreasing, assuming we only consider x ≥ 0. If we consider x < 0, f'(x) would be negative for x < -1, indicating the function is decreasing on the interval (-INF, -1).