First derivative test

In calculus, the first derivative test uses the first derivative of a function to determine whether a given critical point of a function is a local maximum, a local minimum, or neither. If f is a function, then f has a relative maximum at x = c if for all points a near c, f(c) > f(a), and f has a relative minimum at x = c if for all points a near c, f(c) < f(a).

First derivative test

Let f be a differentiable function with f'(c) = 0, then

• If f'(x) changes from positive to negative, then f has a relative maximum at c.
• If f' (x) changes from negative to positive, then f has a relative minimum at c.

Suppose f(x) is continuous at a stationary point x₀.

If f'(x) > 0 on an open interval extending left from x₀ and f'(x) < 0 on an open interval extending right from x₀, then f(x) has a relative maximum (possibly a global maximum) at x₀. If f'(x) < 0 on an open interval extending left from x₀ and f'(x) > 0 on an open interval extending right from x₀, then f(x) has a relative minimum (possibly a global minimum) at x₀.

If f'(x) has the same sign on an open interval extending left from x₀ and on an open interval extending right from x₀, then f(x) has an inflection point at x₀.

This is a method for determining whether an inflection point is a minimum, maximum, or neither.

If the derivative of a function changes sign around a critical point, the function is said to have a local (relative) extremum at that point. If the derivative changes from positive (increasing function) to negative (decreasing function), the function has a local (relative) maximum at the critical point. If, however, the derivative changes from negative (decreasing function) to positive (increasing function), the function has a local (relative) minimum at the critical point. When this technique is used to determine local maximum or minimum function values, it is called the First Derivative Test for Local Extrema. Note that there is no guarantee that the derivative will change signs, and therefore, it is essential to test each interval around a critical point.

Consider the example :

f(x) = x⁴ - 8 x² we have to find all local extrema for the function.

f(x) has critical points at x = -2, 0, 2. Because f'(x) changes from negative to positive around -2 and 2, f has a local minimum at (-2,-16) and (2,-16). Also, f'(x) changes from positive to negative around 0, and hence, f has a local maximum at (0,0).