Find the largest and smallest values of the function y = (x-2) ^ 3 + 4 on the segment [0,3].

y = (x – 2) ^ 3 + 4;

1. Find the derivative of the given function:

y ‘= ((x – 2) ^ 3 + 4)’ = 3 (x – 2) ^ 2;

2. Find the critical points:

3 (x – 2) ^ 2 = 0;

3 (x ^ 2 – 4x + 4) = 0;

x ^ 2 – 4x + 4 = 0;

D = 16 – 4 * 1 * 4 = 0;

x = 4/2 = 2;

3. Find the values of the function at the points and at the ends of the segment:

y (0) = (0 – 2) ^ 3 + 4 = 2;

y (2) = (2 – 2) ^ 3 + 4 = 4;

y (3) = (3 – 2) ^ 3 + 4 = 5;

Answer: min y (0) = 2, max y (3) = 5.