To answer these questions, we need to perform some calculus operations.
The first step is to differentiate the function. The derivative of the function f(x) will measure how f(x) changes as x changes.
We know that the function is f(x) = 2x³ - 24x + 6.
We differentiate it and get f'(x) = 6x² - 24.
Next, we want to find the values of x where f'(x) equals 0, since these locations could be points where the function switches from increasing to decreasing or vice versa. Therefore, solving f'(x) = 0 yields x = -2 and x = 2. These are called the critical points.
We can now apply the First Derivative Test to classify these critical points and thus determine where on the function is increasing or decreasing. We do this by choosing test points in the intervals created by these critical points and checking the sign of f'(x) at those points because the sign of the derivative indicates whether the function is increasing or decreasing.
Here, our critical points split the number line into three intervals, (-∞, -2), (-2, 2), and (2, ∞).
Taking a point from each interval, say -3, 0, and 3, and substituting these into the derivative function, f'(x):
For the interval (-∞, -2), consider x=-3, f'(-3) = 6*(-3)² - 24 = 54 > 0. So, the function is increasing on (-∞, -2).
For the interval (-2, 2), consider x = 0, f'(0) = 6*(0)²-24 = -24 which is less than 0. So, the function is decreasing on this interval, i.e., (-2, 2).
Lastly, for the interval (2, ∞), consider x=3, f'(3) = 6*(3)² - 24 = 54 > 0. So, the function is increasing on (2, ∞).
So, in conclusion:
a) The function is increasing on the intervals (-∞, -2) and (2, ∞).
b) The function is decreasing on the interval (-2, 2).