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 is said to be increasing on an interval if for all where it follows that In a similar fashion, is said to be decreasing on an interval if for all where it follows that
Because the slope of the line tangent to the graph of the function 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 and the first derivative switches sign at , then the function exhbits a local extremum at