Find the largest value of the function y = x ^ 3 + 6x ^ 2 + 19 on the segment [-6; -2].

1. Let’s find the stationary points of the function:

y = x ^ 3 + 6x ^ 2 + 19;

y ‘= 3x ^ 2 + 12x = 3x (x + 4);

y ‘= 0;

3x (x + 4) = 0;

[x = 0;
[x + 4 = 0;

[x = 0 ∉ [-6; -2];
[x = -4 ∈ [-6; -2].

2. Let’s calculate the value of the function at the ends of a given segment and at the extremum point belonging to this segment:

y = x ^ 3 + 6x ^ 2 + 19;

y (-6) = (-6) ^ 3 + 6 * (-6) ^ 2 + 19 = -216 + 216 + 19 = 19;
y (-4) = (-4) ^ 3 + 6 * (-4) ^ 2 + 19 = -64 + 96 + 19 = 51;
y (-2) = (-2) ^ 3 + 6 * (-2) ^ 2 + 19 = -8 + 24 + 19 = 35.
The function takes its greatest value at the maximum point:

y (max) = y (-4) = 51.

Answer: 51.