Understanding Increasing/Decreasing Functions, Concavity, and Extrema

Increasing means the function rises as you move from left to right on the graph, while decreasing means it falls in that direction. Concavity describes the direction of the curve’s bend: a graph that opens upward is concave up, and one that opens downward is concave down.

How Derivatives Reveal Function Behavior

The first derivative gives the slope of the tangent line. When the derivative is positive on an interval, the function is increasing there; when it is negative, the function is decreasing. If the derivative equals zero, the function is locally constant at that point or over that interval.

Concavity and the Role of the Second Derivative

Concavity reflects how the slope itself changes. A positive second derivative means the slope is increasing, producing a concave‑up shape that can hold water. A negative second derivative means the slope is decreasing, producing a concave‑down “umbrella” shape that sheds water. When the second derivative is zero, the point may be an inflection point where the curvature switches.

Inflection Points

An inflection point occurs where the graph changes from concave up to concave down, or vice versa. These points often appear where the second derivative is zero or undefined.

Relative Extrema

A relative maximum (a peak) happens when the function changes from increasing to decreasing, typically where the first derivative crosses zero from positive to negative. A relative minimum (a valley) occurs when the function changes from decreasing to increasing, where the first derivative crosses zero from negative to positive. These are local high or low points, not necessarily the highest or lowest values on the entire graph.

Critical Numbers

Critical numbers are points in the domain where the first derivative is zero or does not exist. They are found by setting the derivative equal to zero and solving, and by identifying where the derivative’s denominator is zero (an undefined slope). Critical numbers are the primary candidates for relative extrema.

Absolute Extrema on Intervals

For a continuous function on a closed interval ([a,b]), the absolute maximum and minimum must occur either at critical numbers inside ((a,b)) or at the endpoints (a) and (b). To locate them, list all critical numbers in the interval, evaluate the function at each critical number and at the endpoints, then compare the values. The largest value is the absolute maximum; the smallest is the absolute minimum. On open intervals, absolute extrema may not exist, but if they do, they must be at critical numbers.

Examples and Applications

Typical examples include determining intervals where a polynomial is increasing or decreasing by analyzing its first derivative, identifying concave‑up and concave‑down regions by checking the sign of the second derivative, and finding critical numbers by solving (f'(x)=0) for a rational function. Absolute extrema are found on a closed interval by evaluating the function at critical numbers and endpoints, and for functions with vertical asymptotes, limits and sign analysis help decide whether extrema exist.