Let \(f\) be the function given by \(f(x)=300x-x^3\). On which of the following intervals is the function \(f\) increasing?

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Apply the first derivative test.

$$ f(x)= 300x-x^3 $$ $$ f'(x)=300-3x^2 \tag*{\scriptsize Differentiate.}$$ $$ 0 = 300-3x^2 \tag*{\scriptsize Find critical numbers.}$$ $$ 0 = 100-x^2 $$ $$ 0 = (10+x)(10-x) $$ $$ x= -10 \text{ and } 10 $$

The table below summarizes testing of the three intervals determined by the critical numbers.

$$ \begin{array}{|c|c|c|c|} \hline \text{Interval} & -\infty \lt x \lt -10 & -10 \lt x \lt 10 & 10 \lt x \lt \infty \\ \hline \text{ Test Value} & x=-11 & x=0 & x= 11 \\ \hline \text{ Sign of }f'(x) & f'(-11) \lt 0 & f'(0) \gt 0 & f'(11) \lt 0 \\ \hline \text{Conclusion} & \text{Decreasing} & \text{Increasing} & \text{Decreasing} \\ \hline \end{array} $$