Final answer:
To find the stationary points of the function f(x), calculate the derivative f'(x) and find the x-values where it equals zero. For f(x) = 2x³ + 3x² - 12x - 7, the stationary points are at x = -2 and x = 1 after factoring the derivative set to zero.
Step-by-step explanation:
To find all the stationary points of the function f(x) = 2x³ + 3x² - 12x - 7, we first need to calculate its derivative f'(x), since stationary points occur where the derivative is zero (i.e., where the slope of the tangent to the curve is horizontal).
The derivative is f'(x) = 6x² + 6x - 12. This is a quadratic equation that can be solved to find the values of x where f'(x) is zero. The quadratic equation is of the form ax² + bx + c = 0.
To solve 6x² + 6x - 12 = 0, we can simplify by dividing through by 6: x² + x - 2 = 0. This equation can be factored into (x + 2)(x - 1) = 0, yielding solutions x = -2 and x = 1.
Next, to determine whether these points are maxima, minima, or saddle points, we would examine the sign of f''(x) or the nature of the curve around these points. In this case, the positions (-2, f(-2)) and (1, f(1)) are the stationary points of f(x).