146k views
0 votes
Point is a relative minimum, relative maximu f(x)=2x³ -3x ² -12x+18;x=2 A

User Xan
by
8.0k points

1 Answer

2 votes

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.

User Arnaud SmartFun
by
7.7k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories