## First Derivative Test for Local Extrema

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.

**Example 1:** If *f(x)* = *x* ^{4} − 8 *x* ^{2}, determine 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).

**Example 2:** If *f(x)* = sin *x* + cos *x* on [0, 2π], determine all local extrema for the function.

*f(x)* has critical points at *x* = π/4 and 5π/4. Because *f′(x)* changes from positive to negative around π/4, *f* has a local maximum at . Also *f′(x)* changes from negative to positive around 5π/4, and hence, *f* has a local minimum at