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 is conclusive, 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 ![]() |
(rules for polynomial differentiation) | ![]() |
Direct. Note that the fact that ![]() ![]() | |
2 | ![]() ![]() |
Fact (1) | Step (1) | Step-fact combination direct | |
3 | All the critical points of ![]() ![]() ![]() |
Fact (2) | Step (2) | Step-fact combination direct | |
4 | The first derivative test is conclusive for ![]() ![]() |
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 ![]() |
(rules for polynomial differentiation, plus quotient rule for differentiation) | ![]() |
Direct. Note that the derivative is nonzero because the function is not constant. | |
2 | ![]() ![]() ![]() ![]() |
Fact (1) | Step (1) | [SHOW MORE] | |
3 | All the critical points of ![]() ![]() ![]() |
Fact (2) | Step (2) | Step-fact combination direct | |
4 | The first derivative test is conclusive for ![]() ![]() |
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.