### Home > CALC > Chapter 5 > Lesson 5.3.2 > Problem 5-123

The 2nd Derivative Test states that if:

If *f* ′(*a*) = 0 AND *f* ′′(*a*) > 0, then *x* = *a* is the location of a local minimum on *f*(*x*).

If *f* ′(*a*) = 0 AND *f* ′′(*a*) < 0, then *x* = *a* is the location of a local maximum on *f*(*x*).

But if *f* ′(*a*) = 0 AND *f* ′′(*a*) = 0, then the test is inconclusive...*f*(*a*) might be a max, min or inflection point.

The 1st Derivative Test states that if:

If *f* '(*a*) = 0 AND *f* ′(*x*) changes from negative to positive values at *x* = *a*, then *x* = *a* is the location of a local minimum on *f*(*x*).

If *f* ′(*a*) = 0 AND *f* ′(*x*) changes from positive to negative values at *x* = *a*, then *x* = *a* is the location of a local maximum on *f*(*x*).

But if *f* ′(*a*) = 0 AND *f* ′(*x*) does not change signs at *x* = *a*, then *x* = *a* is the location of an inflection point on *f*(*x*).

Functions can have an undefined slope (derivative) at a cusp, endpoint, jump, hole or vertical tangent.

Do any of those scenarios match the given one: *g* ′(4) = DNE but *g* ′(*x*) changes from negative to positive at *x* = 4.

Refer to hint in part (b), the 1st Derivative Test.