Find the largest value of f(x) = x³ − 6x² + 9x + 1 on the closed interval [0, 5]. Give the answer as an integer.

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Highest point on an interval

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Find the largest value of $f (x) = x^{3} - 6 x^{2} + 9 x + 1$ on the closed interval $[0, 5]$ . Give the answer as an integer.

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Hint

Compare the values at the critical points inside the interval with the values at the endpoints.

Solution

$f^{'} (x) = 3 x^{2} - 12 x + 9 = 3 (x - 1) (x - 3)$ , so the critical points are $x = 1$ and $x = 3$ .
The values there are $f (1) = 1 - 6 + 9 + 1 = 5$ and $f (3) = 27 - 54 + 27 + 1 = 1$ .
At the endpoints, $f (0) = 1$ and $f (5) = 125 - 150 + 45 + 1 = 21$ .
The largest of these is the endpoint value $21$ .

Answer: $21$