Extrema on an Interval/ First Derivative Test

The First Derivative Test is a method used to determine local extrema of a function by analyzing the sign of its first derivative. If f'(x) changes from positive to negative at a point, there is a local maximum; if it changes from negative to positive, there is a local minimum.

If f'(x) > 0 over an interval, it means that the function f(x) is increasing on that interval.

If f'(x) < 0 over an interval, it means that the function f(x) is decreasing on that interval.

A critical point of a function f(x) is a point x = c in the domain of f where f'(c) = 0 or f'(c) does not exist. Critical points are potential locations for local maxima and minima.

To identify local maxima and minima using a sign chart, determine the intervals where f'(x) is positive or negative. A change from positive to negative indicates a local maximum, while a change from negative to positive indicates a local minimum.

The equation g'(-2) = 0 indicates that x = -2 is a critical point of the function g(x). This point may correspond to a local maximum, local minimum, or an inflection point.

If a function is constant on an interval, it means that the function's value does not change over that interval, which implies that its derivative f'(x) = 0 for all x in that interval.

The first derivative indicates the slope of the function (increasing or decreasing), while concavity is determined by the second derivative. If f''(x) > 0, the function is concave up; if f''(x) < 0, the function is concave down.

An inflection point is a point on the graph of a function where the concavity changes. This occurs when the second derivative f''(x) changes sign.

To determine if a function is increasing or decreasing on a given interval, analyze the sign of the first derivative f'(x). If f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing.