# First derivative test is conclusive for function with algebraic derivative

This article describes a situation, or broad range of situations, where a particular test or criterion isconclusive, i.e., it works as intended to help us determine what we would like to determine.

The test is first derivative test. See more conclusive cases for first derivative test | inconclusive cases for first derivative test

## Contents

## Statement

### Single definition case

For any function of the following type, the first derivative test is always conclusive, i.e., it can be used to definitively determine whether the function has a local extreme value at a given critical point, and if so, what the nature of the local extreme value is:

- Nonconstant polynomial function on the real line, or on an interval or union of finitely many intervals in the real line
- Nonconstant rational function on the real line (whatever subset it's defined on), or on an interval or union of finitely many intervals in its maximum possible domain
- Nonconstant function defined on the real line, or on an interval or union of finitely many intervals in the real line, such that the derivative of the function is a rational function on its domain

Note that we omit constant functions from consideration because the derivative is identically zero and the analysis of local extreme values does not require the use of derivatives.

### Piecewise definition case

- Function that has a piecewise definition by interval in terms of polynomials and rational functions, and is continuous at all transition points between intervals. (Note that if it is not continuous at a transition point, we'll need to use the variation of first derivative test for discontinuous function with one-sided limits)

## Related facts

- First derivative test is conclusive for locally analytic function
- Higher derivative test is conclusive for function with algebraic derivative

## Facts used

- Nonzero polynomial function has finitely many zeros
- Any finite set is a discrete set, and all points in that set are isolated
- First derivative test is conclusive for differentiable function at isolated critical point

## Proof

The proofs in all cases are essentially the same, but we present them separately to make them accessible to people who are not familiar with the more advanced cases.

### Case of polynomial function on the real line

**Given**: A nonconstant polynomial function , a critical point for

**To prove**: The first derivative test is conclusive for at . In other words, has constant sign on the immediate left of , and it has constant sign on the immediate right of .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | The derivative is a nonzero polynomial function. |
(rules for polynomial differentiation) | is a nonconstant polynomial function | Direct. Note that the fact that is nonconstant forces to not be identically zero. | |

2 | is defined everywhere and has finitely many zeros, so has finitely many critical points. | Fact (1) | Step (1) | Step-fact combination direct | |

3 | All the critical points of are isolated critical points. In particular, is an isolated critical point for . | Fact (2) | Step (2) | Step-fact combination direct | |

4 | The first derivative test is conclusive for at . | Fact (3) | Step (3) | Step-fact combination direct |

### Case of rational function on its maximum possible domain

**Given**: A (reduced form, i.e., no common factors between numerator and denominator) nonconstant rational function , a critical point for

**To prove**: The first derivative test is conclusive for at . In other words, has constant sign on the immediate left of , and it has constant sign on the immediate right of .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | The derivative is a nonzero rational function. | (rules for polynomial differentiation, plus quotient rule for differentiation) | is a rational function | Direct. Note that the derivative is nonzero because the function is not constant. | |

2 | is defined everywhere on the domain of and has finitely many zeros, so has finitely many critical points given by the zeros of . | Fact (1) | Step (1) | [SHOW MORE] | |

3 | All the critical points of are isolated critical points. In particular, is an isolated critical point for . | Fact (2) | Step (2) | Step-fact combination direct | |

4 | The first derivative test is conclusive for at . | Fact (3) | Step (3) | Step-fact combination direct |

### Case of function whose derivative is a rational function on its domain

The proof is similar to the rational function case, except that we don't have to do Step (1) of the proof above and can proceed directly from Step (2) onward.