Final answer:
To classify the stationary points of the function y=x³-(3x²)/2-6x+4, first find the first derivative and set it to zero to locate the points. Then use the second derivative to classify each as a maximum, minimum, or inflection point based on its sign at those points.
Step-by-step explanation:
To find and classify all of the stationary points of the function y = x³ - (3x²)/2 - 6x + 4, we first compute the first derivative which reveals the stationary points by setting it to zero. The first derivative is y' = 3x² - 3x - 6. Setting y' to zero yields 3x² - 3x - 6 = 0. This can be factored or solved using the quadratic formula to find the values of x where the function has stationary points.
We then take the second derivative, y'' = 6x - 3, to determine the nature of each stationary point. By substituting the x-values into the second derivative, we can classify each point as a maximum, minimum, or inflection point based on the sign of y''. If y'' is positive at a stationary point, it's a minimum; if negative, it's a maximum; and if zero, it suggests an inflection point, though further analysis may be required.