Why Sign Charts Help with Polynomial Extrema
Finding where a polynomial’s derivative is zero identifies possible locations of local extrema, but it does not classify those locations by itself. A critical number can be a local maximum, a local minimum, or a stationary point that is neither. The signs of the derivative immediately to its left and right distinguish these cases.
Polynomials are particularly suited to this method. Their derivatives are also polynomials, and a nonzero polynomial has only finitely many real roots. Factoring the derivative makes those roots and their sign changes visible. The constant-polynomial case must be treated separately: its derivative is zero everywhere, and every point is both a non-strict local maximum and a non-strict local minimum.
Definitions and the Sign Test
For a polynomial defined on all of \(\mathbb R\), there are no domain breaks to consider. Since polynomials are differentiable everywhere, Fermat’s theorem says that any local maximum or minimum point \(c\) must satisfy \(f'(c)=0\). Thus, for a nonconstant polynomial, its critical numbers are precisely the real solutions of \(f'(x)=0\).
To classify one of these numbers, compare the derivative’s signs on the intervals immediately to its left and right. A positive derivative means the function is increasing on that interval; a negative derivative means it is decreasing. The First Derivative Test therefore gives the following sign-change rules.
| Sign of \(f'\) before \(c\) | Sign of \(f'\) after \(c\) | Conclusion at \(c\) |
|---|---|---|
| Positive | Negative | Strict local maximum |
| Negative | Positive | Strict local minimum |
| Positive | Positive | No local extremum |
| Negative | Negative | No local extremum |
For a nonconstant polynomial, each zero of \(f'\) is isolated: there is a small interval around it containing no other zero of \(f'\). On either side, the derivative is nonzero and its sign is constant. This is why a sign chart can classify every critical number. If the derivative factors, multiplicities often make the signs especially easy to read: a factor \((x-c)^m\) changes sign across \(c\) exactly when \(m\) is odd, provided the other factors are nonzero there.
Why the Sign Change Determines the Extrema
Here is a justification of the test in the polynomial setting. Let \(f\) be a nonconstant polynomial and let \(c\) be a zero of \(f'\). Choose a small interval around \(c\) containing no other zero of \(f'\). On each side, \(f'\) has one fixed, nonzero sign. The mean value theorem says that if \(f'>0\) throughout an interval, then \(f\) is strictly increasing there: for any \(u<v\), there is a \(\xi\in(u,v)\) such that \(f(v)-f(u)=f'(\xi)(v-u)>0\). If \(f'<0\), the same equation shows \(f(v)<f(u)\).
Suppose first that \(f'\) is positive to the left of \(c\) and negative to the right. For any nearby \(x<c\), the mean value theorem on \([x,c]\) gives \(f(c)>f(x)\), because the intermediate point lies strictly to the left of \(c\). For any nearby \(y>c\), applying it on \([c,y]\) gives \(f(y)<f(c)\). Thus \(c\) is a strict local maximum. With the signs reversed, the same comparisons show that \(c\) is a strict local minimum. If the signs agree, the function is strictly increasing on both sides or strictly decreasing on both sides, so nearby values occur both below and above \(f(c)\); \(c\) is neither kind of local extremum.
Worked Example: A Repeated Critical Number That Is Not an Extremum
Worked Example: A Repeated Critical Number That Is Not an Extremum
Classify all critical numbers of \(f(x)=\frac{3}{4}x^4-\frac{9}{2}x^2-6x\), and find the function values at any extrema.
Differentiate and factor:
The critical numbers are \(x=-1\) and \(x=2\). To find signs, note that \(3\) and \((x+1)^2\) are positive except that the squared factor is zero at \(-1\). Thus away from the critical numbers, the sign is determined by \(x-2\), which is negative for \(x<2\) and positive for \(x>2\).
At \(-1\), the derivative is negative on both sides, so the function is decreasing through that point. It is not a local extremum. At \(2\), the derivative changes from negative to positive, so \(x=2\) is a strict local minimum. Its value is
The minimum point is \((2,-18)\). Check the derivative by differentiating the displayed polynomial: \(\frac{d}{dx}(\frac{3}{4}x^4)=3x^3\), \(\frac{d}{dx}(-\frac{9}{2}x^2)=-9x\), and \(\frac{d}{dx}(-6x)=-6\). These terms give \(3x^3-9x-6=3(x+1)^2(x-2)\), as required.
Worked Example: Several Local Extrema
Worked Example: Several Local Extrema
Find and classify every critical number of \(g(x)=\frac{3}{4}x^4-3x^3-9x^2+24x\). Give the coordinates of the local extrema.
Differentiate and factor the result:
The critical numbers are \(-2\), \(1\), and \(4\). Check the sign of the three linear factors on each interval:
At \(-2\), the sign changes from negative to positive, so there is a strict local minimum. At \(1\), it changes from positive to negative, so there is a strict local maximum. At \(4\), it changes from negative to positive, so there is another strict local minimum.
Evaluate the polynomial at each critical number:
The local minima are \((-2,-48)\) and \((4,-48)\), and the local maximum is \((1,\frac{51}{4})\). To check the derivative, expanding the factored form gives \(3(x+2)(x-1)(x-4)=3(x^3-3x^2-6x+8)=3x^3-9x^2-18x+24\), which is exactly the derivative computed from \(g\).
Worked Example: The Constant-Polynomial Exception
Worked Example: The Constant-Polynomial Exception
Classify the local extrema of \(h(x)=7\) on \(\mathbb R\).
Its derivative is \(h'(x)=0\) for every real \(x\). For any \(c\in\mathbb R\) and any \(\delta>0\), every \(x\) satisfying \(|x-c|<\delta\) has \(h(x)=7=h(c)\). Consequently, both inequalities \(h(x)\leq h(c)\) and \(h(x)\geq h(c)\) hold throughout every such neighborhood. Every real number is therefore both a non-strict local maximum point and a non-strict local minimum point. No point is a strict local maximum or strict local minimum, because nearby values are equal, not strictly smaller or larger.
This example is why the no-sign-change conclusion must be stated for a critical number of a nonconstant polynomial: here the derivative is zero everywhere, so there are no adjacent intervals on which it has a positive or negative sign.
A Reliable Procedure and a Common Mistake
For a nonconstant polynomial, a complete local-extrema analysis follows a short sequence. First differentiate and factor \(f'\). Next solve \(f'(x)=0\) for all real critical numbers. Then mark them in order on a number line, determine the derivative’s sign on each resulting interval, and read each sign change. Finally, evaluate \(f\) at the critical numbers classified as extrema if coordinates are wanted. A sign chart classifies local behavior; it does not compare extrema at distant points.
For a nonconstant polynomial, solve \(f'(x)=0\); the derivative exists everywhere.
Use the factored derivative to determine its sign on each interval between consecutive critical numbers.
Positive to negative means local maximum; negative to positive means local minimum; unchanged nonzero sign means neither.
Substitute each classified critical number into \(f\) to obtain the extremum’s coordinates.
A common mistake is to label every solution of \(f'(x)=0\) as a local extremum. For example, in the first worked example, \(f'(-1)=0\), but \(f'\) is negative on both sides of \(-1\). Calling \(x=-1\) a local minimum merely because it is a critical number would be wrong: the function decreases both before and after it. The sign chart corrects the classification. A second, related mistake is to apply the no-sign-change rule to the constant polynomial \(h(x)=7\). That rule concerns a nonconstant polynomial with a nonzero derivative sign on either side; the constant example has every point as both kinds of non-strict local extremum.
Check Your Understanding
Use derivative signs, rather than the equation \(f'(x)=0\) alone, to answer these questions.
- For \(p'(x)=(x-3)^2(x+1)\), what are the critical numbers, and which are local extrema of \(p\)?
- If a nonconstant polynomial’s derivative changes from positive to negative at \(c\), what kind of local extremum occurs there?
- Why does a critical number with the same nonzero derivative sign on both sides fail to be a local extremum?
- For a constant polynomial, how are its points classified under the non-strict definitions of local maximum and local minimum?
- What additional calculation is needed to give the coordinates of a local extremum after its critical number has been classified?
