Find the largest and smallest values of the function y = 2x ^ 3 + 3x ^ 2-12x-1 on the segment [-1; 2]

We calculate the derivative and find its zeros (critical points of the original function), we get:

y ‘(x) = x² + x – 2 = 0.

By Vieta’s theorem, we get the roots:

x = -2,

x = 1.

The point x = -2 is the maximum point of this function, because the derivative changes its sign from “plus” to “minus”, therefore, y (-2) = 19.

The point x = 1 is the minimum point of the original function, since the derivative at this point changes its sign from “minus” to “plus”, therefore, we have a minimum at (1) = -8.

Answer: ymax = 19, ymin = -8.