Find the smallest value of the function y = x ^ 3 – 18x ^ 2 + 17 on the segment [6; 18].

Find critical points:

y ‘= 3x ^ 2 -36x;

3x ^ 2 -36x = 0;

x (3x – 36) = 0;

x1 = 0;

Point x1 does not belong to the line segment.

x2 = 12;

Point x2 is critical.

Two more critical points will be the end points of the line segment:

x3 = 6;

x4 = 18.

We find the value of the function at critical points:

y2 = x2 ^ 3 – 18×2 ^ 2 + 17 = 12 ^ 3 – 18 * 12 ^ 2 +17 = -847;

y3 = x3 ^ 3 – 18×3 ^ 2 + 17 = 6 ^ 3 – 18 * 6 ^ 2 + 17 = -415;

y4 = x2 ^ 3 – 18×4 ^ 2 + 17 = 18 ^ 3 – 18 * 18 ^ 2 + 17 = 17.

Answer. The function takes the smallest value on the segment at the point x2 = 12: y (12) = -847.