The graph of the function crosses the x-axis at x = 0, x = -1, and x = -3 and therefore, f(x) is positive in one of the intervals of the x-intercept
The correct options are;
- f(x) is negative in the intervals (-∞, -3) and (-1, 0)
- f(x) is increasing on the intervals (-∞, -2) and (-0.5, ∞)
- f(x) is positive on the intervals (-3, -1) and (0, ∞)
- f(x) is decreasing on the interval -2 ≤ x ≤ -0.5
Reason:
The given function is presented as follows;
f(x) = x³ + 4·x² + 3·x
The graph of the above function is created with MS Excel
From the function, we have; f(x) = x³ + 4·x² + 3·x = x·(x + 3)·(x + 1)
From the graph, we have;
f(x) is negative in the interval -∞ < x < -3, and -1 ≤ x ≤ 0 which gives;
- f(x) is negative in the intervals (-∞, -3) and (-1, 0)
The value of f(x) is increasing on the interval -∞ < x ≤ -2 and also in the interval -0.5 ≤ x ≤ ∞, therefore;
- f(x) is increasing on the intervals (-∞, -2) and (-0.5, ∞)
The value of the function f(x) is positive on the interval -3 ≤ x ≤ -1, and also positive in the interval 0 ≤ x < ∞
Therefore;
- f(x) is positive on the intervals (-3, -1) and (0, ∞)
The value of f(x) is decreasing for values of x between -2 ≤ x ≤ -0.5
Therefore;
- f(x) is decreasing on the interval -2 ≤ x ≤ -0.5
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