Find the largest and smallest value of the function f (x) = x ^ 3-2x ^ 2 + x + 3 on the segment [0; 3/2]

Answer: the largest value of the function at x = 3/2;
the smallest – at x = 0 and x = 1.
Explanation: we find OOF: x – any number
Find the derivative of the function: f` (x) = 3x ^ 2-4x + 1 = 0 (equate to zero)
We solve the resulting quadratic equation: x1 = 1, x2 = 1/3
We find the value of the function at these points and on the boundaries of the segment:
f (x) = x ^ 3-2x ^ 2 + x + 3
f (0) = 0 ^ 3-2 * 0 ^ 2 + 0 + 3 = 3
f (3/2) = (3/2) ^ 3-2 * (3/2) ^ 2 + 3/2 + 3 = 27/8
f (1) = 1 ^ 3-2 * 1 ^ 2 + 1 + 3 = 3
f (1/3) = (1/3) ^ 3-2 * (1/3) ^ 2 + 1/3 + 3 = 85/27
Compare fractions at x = 1/3 and x = 3/2: 85 * 8/27 = 680/213, 27 * 27/8 = 729/216.