Find the smallest and largest value of the function y = -2 (x-1) ^ 2 on the segment [-1,2].

1. Let’s find the first derivative of the given function:

y ‘= (-2 * (x – 1) ^ 2)’ = -4 * (x – 1).

2. Let us equate this derivative to zero and find the critical points:

-4 * (x – 1) = 0;

-4x + 4 = 0;

-4x = -4;

x = -4: (-4);

x = 1.

3. Find the value of the function at this point and at the ends of the specified segment [-1; 2]:

y (1) = -2 * (1 – 1) ^ 2 = -2 * 0 = 0;

y (-1) = -2 * (-1 – 1) ^ 2 = -2 * (-2) ^ 2 = -2 * 4 = -8;

y (2) = -2 * (2 – 1) ^ 2 = -2 * 1 = -2.

The largest value of the function at the point x = 1, the smallest value of the function at the point x = -1.

Answer: fmax = 0, fmin = -8.