Final answer:
To find the local minimum of the cubic function, one must calculate the first derivative to determine critical points and then use the second derivative to identify if these points are minima or maxima.
Step-by-step explanation:
The function f(x)=-2x³ + 36x² -162x+3 has one local minimum and one local maximum. To find the local minimum, we need to take the derivative of the function to find the critical points. The derivative of the function is f'(x)=-6x² + 72x -162. This is a quadratic equation that can be solved for x using the quadratic formula, x = (-b ± √(b² - 4ac))/(2a). Setting the derivative equal to zero will give us the critical points. The second derivative of the function, f''(x)=-12x + 72, will tell us whether these critical points are minima or maxima. If the second derivative is positive at a critical point, then that point is a minimum. If it is negative, then that point is a maximum.