Final answer:
To find the intervals of increase and decrease for the function f(x) = x³ + 12x² + 21x + 1, we need to find the critical points.
Step-by-step explanation:
To determine the intervals where the function f(x) = x³ + 12x² + 21x + 1 is increasing or decreasing, we need to find the critical points. These occur when the derivative of the function equals zero or does not exist. Differentiating f(x) with respect to x gives f'(x) = 3x² + 24x + 21.
Setting f'(x) = 0 and solving for x, we get x = -7 and x = -1.
By testing values for x in the intervals (-∞, -7), (-7, -1), and (-1, ∞), we find that f(x) is increasing in the interval (-∞, -7) and (-1, ∞), and decreasing in the interval (-7, -1).