Question

Find the critical points of the function \( f(x) = 2x^3 - 3x^2 - 12x + 15 \):

Hint:

To find critical points, set the first derivative of the function equal to zero and solve for \( x \).

Answer 1

Step 1: Finding the first derivative.
To find the critical points, we first find the first derivative of \( f(x) = 2x^3 - 3x^2 - 12x + 15 \): \[ f'(x) = 6x^2 - 6x - 12 \] Step 2: Setting the first derivative equal to zero.
Set \( f'(x) = 0 \) to find the critical points: \[ 6x^2 - 6x - 12 = 0 \] Simplifying: \[ x^2 - x - 2 = 0 \] Step 3: Solving the quadratic equation.
Factor the quadratic equation: \[ (x - 2)(x + 1) = 0 \] Thus, \( x = 2 \) and \( x = -1 \) are the critical points. Step 4: Conclusion.
The critical points of the function are \( x = 2 \) and \( x = -1 \).