# Find the largest and smallest values of the function y = x ^ 3-9x ^ 2 + 24x-18 on the segment {0; 3}.

June 30, 2021 | education

| Let’s find the derivative of the function:

y ‘= 3x ^ 2 – 18x + 24.

Find the extremum points of the function from the equation:

3x ^ 2 – 18x + 24 = 0;

x ^ 2 – 6x + 8 = 0;

By the Vieta converse theorem, x1 = 2; x2 = 4.

x2 = 4 is outside the segment of interest to us, therefore, we will not consider it further.

Find the values of the function at the ends of the segment and at the point х1 = 2:

y (0) = 0 ^ 3 – 9 * 0 ^ 2 + 24 * 0 – 18 = – 18;

y (2) = 2 ^ 3 – 9 * 2 ^ 2 + 24 * 2 – 18 = 8 – 36 + 48 – 18 = 2;

y (3) = 3 ^ 3 – 9 * 3 ^ 2 + 24 * 3 – 18 = 27 – 81 + 72 – 18 = 0.

So, the largest value of the function on a given interval is 2, the smallest is 18.

Answer: the largest value of the function is 2, the smallest is 18.

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