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The maximum value of $f(x)=x^3-9x^2+24x+5$ in the interval $[1,6]$ is

1. 21
2. 25
3. 41
4. 46

Correct answer: 41

Solution

Differentiate to get $f'(x)=3x^2-18x+24=3(x-2)(x-4)$, so the critical points are $x=2$ and $x=4$. Evaluating $f(x)$ at $x=1,2,4,6$ gives $21,33,25,$ and $41$ respectively, so the maximum is $41$.

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