Increasing / Decreasing Functions

Increasing and decreasing are properties in real analysis that give a sense of the behavior of functions over certain intervals. For differentiable functions, if the derivative of a function is positive on an interval, then it is known to be increasing while the opposite is true if the function's derivative is negative.

A function $f$ is said to be increasing on an interval $I$ if for all $a, b \in I$ where $a \leq b,$ it follows that $f(a) \leq f(b).$ In a similar fashion, $f$ is said to be decreasing on an interval $I$ if for all $a, b \in I$ where $a \leq b,$ it follows that $f(a) \geq f(b).$

Because the slope of the line tangent to the graph of the function $y = f(x)$ is positive when the derivative is positive, we can deduce that a function is increasing on intervals where its derivative is positive. Similarly, when a function's derivative is negative on an interval, it must be that the function is decreasing on that interval.

By partitioning the number line into intervals on which a function's derivative is positive and negative, the behavior of a function can be determined based on the sign of the derivative on each subinterval. Furthermore, if it is known that $f'(c) = 0$ and the first derivative switches sign at $x = c$, then the function exhbits a local extremum at $x = c.$