Consider the function . In which interval is the graph increasing?

To determine the intervals where the function is increasing, we can find the derivative . The derivative is calculated as follows:

This derivative is always non-negative since squaring a number yields a positive result, and thus for all . The derivative equals zero specifically at , which is a critical point. Since the derivative is positive for all except at , we conclude that is increasing on the intervals and . Therefore, the function is increasing in the interval .

So the correct choice is .

For further exploration, it’s fascinating to note that cubic functions like this one often exhibit a general “S-shaped” curve. They usually have one local maximum and one minimum or can be monotonic, depending on the coefficients. In this instance, the point acts as an inflection point, transforming the increase in a fun way!

Also, when working with polynomial functions, remember to check corresponding intervals of the derivative. Never forget that the zeros in the derivative can indicate potential maximums and minimums. Given that you’re dealing with squares, keep an eye out for alterations in behavior around those critical points; they’re often where the real changes happen!