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 a 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 \(x = c.\)