When $f(x)$ is said to be increasing at $x_0$, it means that the slope of the function $f(x)$ at $x_0$ is positive. Mathematically, this is characterized by the derivative $f'(x_0)$ being positive:

$f'(x_0) > 0$

This implies that as $x$ moves slightly to the right of $x_0$ ($x > x_0$), the function values $f(x)$ increase, i.e., $f(x) > f(x_0)$. Similarly, for values of $x$ slightly to the left of $x_0$ ($x < x_0$), the function values $f(x)$ are less than $f(x_0)$.

In more intuitive terms:

• Graphically: The curve of $f(x)$ is sloping upwards at $x_0$.
• Numerically: The rate of change of $f(x)$ with respect to $x$ at $x_0$ is positive.

Would you like a deeper explanation, visualizations, or examples?

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Expanding Questions:

1. How is the derivative used to determine intervals of increase and decrease for a function?
2. What is the difference between a function being increasing and being strictly increasing?
3. How does the second derivative affect the behavior of $f(x)$ when it is increasing?
4. Can a function be increasing at a point where the derivative does not exist? If so, how?
5. How would you test whether a function is increasing over an interval instead of just at a point?

Tip:

To identify increasing or decreasing behavior, always calculate and analyze the first derivative.