Find the largest and smallest value of the function f (x) = x ^ 3-6x ^ 2 + 9 on the segment [-2; 2].

Find the derivative of the function:

(f (x)) ‘= (x ^ 3 – 6x ^ 2 + 9)’ = 3 * x ^ 2 -12 * x.

Let’s equate it to zero:

3 * x ^ 2 – 12 * x = 0

x * (x – 4) = 0

x1 = 0, x2 = 4

The root x2 does not belong to the given segment.

Find the values of the function at the ends of the segment and at the extremum point.

y (-2) = (-2) ^ 3 – 6 * (-2) ^ 2 + 9 = -8 – 24 + 9 = -23

y (2) = 2 ^ 3 – 6 * 2 ^ 2 + 9 = 8 – 24 + 9 = -7

y (0) = 9.

Answer: the maximum value of the function on the segment 9, the minimum -23.