Question Detail

To find the minimum (or maximum) value of a function, we follow these steps:1. Find the first derivative of the function, dy/dx.2. Set dy/dx = 0 to find the critical points.3. Use the second derivative test (d2y/dx2) to determine if each critical point is a local maximum or minimum. If d2y/dx2 > 0, it's a local minimum. If d2y/dx2 < 0, it's a local maximum.4. Evaluate the original function y at the local minimum point(s).Given y = x3 - 3x2 + 1.1. dy/dx = 3x2 - 6x.2. Set dy/dx = 0: 3x2 - 6x = 0 => 3x(x - 2) = 0. The critical points are x = 0 and x = 2.3. Find the second derivative: d2y/dx2 = 6x - 6. - At x = 0: d2y/dx2 = 6(0) - 6 = -6. Since -6 < 0, x=0 corresponds to a local maximum. - At x = 2: d2y/dx2 = 6(2) - 6 = 12 - 6 = 6. Since 6 > 0, x=2 corresponds to a local minimum.4. Evaluate y at x = 2: y(2) = (2)3 - 3(2)2 + 1 = 8 - 3(4) + 1 = 8 - 12 + 1 = -3.Thus, the minimum value of y is -3.This matches option C.