# Second derivative test

The second derivative test is used to determine whether a function has a relative minimum or maximum at a critical point. A critical point is a point at which the first derivative of a function, f'(x), equals 0. It can also be used to test the concavity of a function and potentially identify inflection points (points where the curve changes from being concave down to concave up, or vice versa).

The second derivative is often denoted as f''(x), where the first derivative is denoted as f'(x).

### Testing for extrema

- If f''(x) is positive at a critical point, the function has a relative maximum at that point.
- If f''(x) is negative at a critical point, the function has a relative minimum at that point.
- If f''(x) is 0 at a critical point, then the test is inconclusive.

### Testing for concavity

- If f''(x) is positive, then the graph of the function, f(x), is concave up.
- If f''(x) is negative, then the graph of the function, f(x) is concave down.

See also first derivative test, derivatives.