Final answer:
The coordinates of the stationary points on the curve y = x³ - 6x² are (0,0) and (4,-32). The point (0,0) is a maximum, and the point (4,-32) is a minimum.
Step-by-step explanation:
To find the coordinates of the stationary points, we need to find the critical points of the curve. These occur where the derivative of the curve is equal to zero or undefined. We can find the derivative of the curve by taking the derivative of each term separately. Taking the derivative of y = x³ - 6x² yields dy/dx = 3x² - 12x. Setting this derivative equal to zero, we get 3x² - 12x = 0. Factoring out x, we get x(3x - 12) = 0. So x = 0 or x = 4. We can substitute these values back into the original equation to find the corresponding y-coordinates. When x = 0, y = (0)³ - 6(0)² = 0. When x = 4, y = (4)³ - 6(4)² = -32. Therefore, the coordinates of the stationary points are (0,0) and (4,-32).
To determine if each point is a maximum or minimum, we can use the second derivative test. Taking the second derivative of the curve, we get d²y/dx² = 6x - 12. Evaluating this at each critical point, we find that at x = 0, d²y/dx² = -12, which is negative. This means that the point (0,0) is a maximum. At x = 4, d²y/dx² = 6(4) - 12 = 12, which is positive. This means that the point (4,-32) is a minimum.