Which statements are true about the function f(x) = x^3 + 2x^2?

To determine the truth of the statements regarding the function ( f(x) = x^3 + 2x^2 ), let’s analyze some key characteristics of the function.

First, we can examine the function by factoring it:

[

f(x) = x^2(x + 2)

]

This reveals that ( f(x) ) has a critical point where ( x^2 ) becomes zero, meaning ( f(x) ) has a double root at ( x = 0 ) and a single root at ( x = -2 ). Evaluating the behavior of the function at these critical points and as ( x ) approaches positive and negative infinity provides insight into its conditions.

Next, we'll look at the derivative of the function to find the turning points and analyze increasing and decreasing intervals:

[

f'(x) = 3x^2 + 4x

]

Factoring this derivative gives:

[

f'(x) = x(3x + 4)

]

Setting the derivative to zero allows us to find critical points:

[

x(3x + 4) = 0 \implies x = 0 \text{ or } x =