### Home > CALC > Chapter 5 > Lesson 5.1.4 > Problem 5-42

When *f* '(*a*) = 0, *f*(*x*) could be at a local max, a local min, or neither (it might be a point of inflection).

The 1st and 2nd derivative tests are ways to tell what is going on at *x* = *a*.

Directions: Evaluate *f* '(*x*) at two points, one to the left of *x* = *a* and the other to the right of *x* = *a*.

☛ If the sign changes from negative to positive, then *f*(*x*) changes from decreasing to increasing and *f*(*a*) is a local min.

☛ If the sign changes from positive to negative, then *f*(*x*) changes from increasing to decreasing and *f*(*a*) is a local max.

☛ If the sign does not change, then *f*(*x*) is either increasing or decreasing (depending on the sign).

Directions: Evaluate *f* "(*a*)

☛ If *f* "(*a*) > 0, then *f*(*x*) is concave up at *x* = *a*, and *f*(*a*) is a local min.

☛ If *f* "(*a*) < 0, then *f*(*x*) is concave down at *x* = *a*, and *f*(*a*) is a local max.

☛ If *f* "(*a*) = 0, then this test is inconclusive.