At x=2, the first derivative of the function f(x) = 2x³ - 3x² - 12x + 18 is zero and the second derivative is positive, indicating that x=2 is a relative minimum, which correlates with stable equilibrium.
To determine whether a point is a relative minimum or relative maximum for the function f(x) = 2x³ - 3x² - 12x + 18 at x=2, we must evaluate the first and second derivatives of the function at this point. If the first derivative equals zero and the second derivative is positive at x=2, it indicates a relative minimum, signaling stable equilibrium. Conversely, if the second derivative is negative, it signifies a relative maximum, indicating unstable equilibrium.
To find the first derivative, we differentiate: f'(x) = 6x² - 6x - 12.
To find the second derivative, we differentiate again: f''(x) = 12x - 6.
Let's evaluate both derivatives at x=2:
- f'(2) = 6(2)² - 6(2) - 12 = 24 - 12 - 12 = 0.
- f''(2) = 12(2) - 6 = 24 - 6 = 18, which is positive.
Since the first derivative is zero and the second derivative is positive at x=2, this point is a relative minimum for the function.