Find the largest and smallest value of the function y = x²-6x + 13 on the interval [0; 6].

1) Find on this segment the critical points f ′ (x) = 0. We get:

f ′ (x) = 2 * x – 6;

f ′ (x) = 0;

2 * x – 6 = 0;

2 * x = 0 + 6;

2 * x = 6;

x = 6: 2;

x = 3;

2) number 3 belongs to the interval 0 ≤ x ≤ 6;

3) We calculate the values of the function at the critical point and at the ends of the interval:

f (3) = 3 ^ 2 – 6 * 3 + 13 = 9 – 18 + 13 = 4;

f (0) = 0 ^ 2 – 6 * 0 + 13 = 0 + 13 = 13;

f (6) = 6 ^ 2 – 6 * 6 + 13 = 36 – 36 + 13 = 13;

4) From the calculated values, select the largest value:

f (0) = f (6) = 13.

Answer: the largest value of the function f (0) = 13, the smallest value f (3) = 4.