Final answer:
To determine the intervals of increase for the function f(x) = 3x⁴ - 4x³ - 12x² + 5, we need to find its derivative, set it to zero, solve for x to get the critical points, and test intervals around these points to see where the derivative is positive.
Step-by-step explanation:
To find the intervals in which the function f(x) = 3x⁴ − 4x³ − 12x² + 5 is strictly increasing, we first need to find the derivative of the function, which represents the slope of the tangent line at any given point on the function. We will look for where this derivative is greater than zero, as this indicates regions where the function is increasing.
The derivative of the function, f'(x), is found using the power rule:
- Take the derivative: f'(x) = 12x³ - 12x² - 24x.
- Find the critical points by setting f'(x) to zero and solving for x: 0 = 12x³ - 12x² - 24x.
- Factor out the common terms: 0 = 12x(x² - x - 2).
- Find the roots of the equation: x = 0, x = -1, and x = 2.
- Use the critical points to test the intervals on either side to see if f'(x) > 0: (-∞, -1), (-1, 0), (0, 2), (2, +∞).
- Choose the intervals where f'(x) is positive.
By testing values in those intervals, we would find out which intervals make f'(x) positive and therefore determine where f(x) is strictly increasing.