Find the extremum points of the function and determine their character y = (x + 1) ^ 3 (3-x).

1. Find the critical points of the function:

y = (x + 1) ^ 3 (3 – x);

y ‘= – (x + 1) ^ 3 + 3 (x + 1) ^ 2 (3 – x) = (x + 1) ^ 2 (3 (3 – x) – (x + 1)) = (x + 1) ^ 2 (9 – 3x – x – 1) = (x + 1) ^ 2 (8 – 4x) = -4 (x + 1) ^ 2 (x – 2);

y ‘= 0;
[x + 1 = 0;
[x – 2 = 0;
[x = -1;
[x = 2.
2. Intervals of monotony:

a) x ∈ (-∞; -1), y ‘> 0, the function is increasing;
b) x ∈ (-1; 2), y ‘> 0, the function is increasing;
c) x ∈ (2; ∞), y ‘<0, the function is decreasing.
x = 2 – maximum point.

Answer. The function has a single extreme point – a maximum point: x = 2.