Critical Numbers and Critical Points

Core Definition

Let $f$ be a function defined on an open interval containing $c$ . A number $c$ in the domain of $f$ ( $c \in \text{dom}(f)$ ) is called a critical number (or critical value of the independent variable) if:

f'(c) = 0 \quad \text{or} \quad f'(c) \text{ does not exist (DNE)}

The ordered pair

(c, f(c))

on the graph of

f

is called a critical point.

Concept Explanation

1. The Anatomy of a Critical Number

Locating the critical numbers of a function is the foundational first step of differential optimization. Every local peak, local valley, or sharp turning point must occur at a critical number.

Critical numbers manifest in two distinct mathematical ways:

Stationary Points ( $f'(c) = 0$ ): The tangent line to the curve at $x = c$ is perfectly horizontal. The instantaneous rate of change temporarily halts. Examples include smooth peaks, smooth troughs, and horizontal inflection plateaus (such as $f(x) = x^3$ at $x = 0$ ).
Singular Points ( $f'(c) \text{ DNE}$ ): The derivative fails to exist at $x = c$ , yet $c$ remains a valid point within the function's domain. Examples include:
- Sharp Corners: Left-hand and right-hand slopes approach different finite values (e.g., $f(x) = |x|$ at $x = 0$ ).
- Sharp Cusps: Slopes approach infinite values of opposite signs (e.g., $f(x) = x^{2/3}$ at $x = 0$ , where $f'(x) = \frac{2}{3x^{1/3}}$ ).
- Vertical Tangents: Slopes approach $+\infty$ or $-\infty$ from both sides (e.g., $f(x) = x^{1/3}$ at $x = 0$ ).

2. Fermat's Theorem and the Converse Trap

Fermat's Theorem on Local Extrema states:

If $f$ has a local extremum (local maximum or local minimum) at $c$ , and if $f'(c)$ exists, then $f'(c) = 0$ .

The Critical Number Converse Trap

Fermat's Theorem guarantees that local extrema only occur at critical numbers. However, the converse is FALSE: A critical number does NOT guarantee the existence of a local extremum!

Counterexample: Consider $f(x) = x^3$ . Its derivative is $f'(x) = 3x^2$ . Setting $f'(x) = 0$ yields $x = 0$ , so $x = 0$ is a valid critical number. However, $f(x)$ is strictly increasing across all real numbers ( $x^3 < 0$ for $x < 0$ , and $x^3 > 0$ for $x > 0$ ). The point $(0, 0)$ is an inflection terrace, neither a local maximum nor a local minimum.
To determine whether a critical number actually generates an extremum, one must apply the First Derivative Test or Second Derivative Test.

3. The Crucial Domain Restriction Rule

A number $c$ can only be classified as a critical number if it belongs to the domain of the original function $f$ :

c \in \text{dom}(f)

The Rational Function Trap: Consider $f(x) = \frac{1}{x}$ . Its derivative is $f'(x) = -\frac{1}{x^2}$ . At $x = 0$ , the derivative is undefined. However, $x = 0$ is NOT a critical number because $x = 0$ is not in the domain of $f(x)$ (it is a vertical asymptote).
The Logarithmic Function Trap: For $f(x) = \ln(x)$ , $f'(x) = \frac{1}{x}$ . While $\frac{1}{x}$ is undefined at $x = 0$ , $0 \notin (0, \infty)$ , so $x = 0$ is not a critical number.

Engineering Applications

Mechanical Cam and Follower Profiling: In internal combustion engines and automated packaging machines, cam lobes are engineered with specific lift profiles $y(\theta)$ . Critical points where $\frac{dy}{d\theta} = 0$ define the valve dwell angles—instants of zero velocity where the intake or exhaust valve remains fully open to maximize airflow.
Electrical Circuit Transient Response: In RLC filter circuits, the instantaneous capacitor voltage $v(t)$ following a switching transient contains critical numbers where $\frac{dv}{dt} = 0$ . Electrical engineers calculate these critical points to determine the exact peak overshoot voltage and prevent dielectric insulation breakdown in semiconductor components.
Structural Load Deflection: In civil structural beams, points of zero slope in the deflection curve $v(x)$ identify the maximum beam sag under distributed dead weight, dictating where reinforcement tendons must be anchored.

Example Problems

Example 1: Critical Numbers of a Polynomial Function

Find all critical numbers of the polynomial function:

f(x) = 2x^3 - 9x^2 + 12x - 5

Solution

Identify the Domain: As a polynomial, the domain of $f(x)$ is all real numbers, $(-\infty, \infty)$ .
Differentiate: $f'(x) = 6x^2 - 18x + 12$
Solve for Stationary Points ( $f'(x) = 0$ ): Factor out the greatest common factor: $6(x^2 - 3x + 2) = 0$ $6(x - 1)(x - 2) = 0$ Setting each linear factor to zero yields: $x = 1 \quad \text{and} \quad x = 2$
Check Singular Points ( $f'(x) \text{ DNE}$ ): Since $f'(x)$ is a polynomial, it is defined everywhere; there are no singular points.
Conclusion: The critical numbers are $x = 1$ and $x = 2$ .

Example 2: Critical Numbers with Singular Points (Cusps)

Find all critical numbers of the function:

g(x) = 5 - (x - 3)^{2/3}

Solution

Identify the Domain: The expression $(x - 3)^{2/3} = \left(\sqrt[3]{x - 3}\right)^2$ involves an odd root (cube root), which is defined for all real numbers. Thus, $\text{dom}(g) = (-\infty, \infty)$ .
Differentiate: Using the Power Rule and Chain Rule: $g'(x) = -\frac{2}{3}(x - 3)^{-1/3} \cdot \frac{d}{dx}[x - 3] = -\frac{2}{3(x - 3)^{1/3}}$
Solve for Stationary Points ( $g'(x) = 0$ ): A fraction equals zero only when its numerator is zero. The numerator is $-2 \neq 0$ . Therefore, $g'(x) \neq 0$ for all $x$ .
Solve for Singular Points ( $g'(x) \text{ DNE}$ ): The derivative is undefined when the denominator equals zero: $3(x - 3)^{1/3} = 0 \implies x - 3 = 0 \implies x = 3$ Because $x = 3$ is in the domain of $g(x)$ ( $g(3) = 5$ ), $x = 3$ is a valid critical number.
Conclusion: The function has exactly one critical number at $x = 3$ (representing a sharp cusp peak on the graph).

Example 3: Rational Function and the Domain Trap

Find all critical numbers of:

h(x) = \frac{x^2}{x - 4}

Solution

Identify the Domain: The denominator cannot be zero: $x - 4 \neq 0 \implies x \neq 4$ . Domain: $(-\infty, 4) \cup (4, \infty)$ .
Differentiate via Quotient Rule: $h'(x) = \frac{\frac{d}{dx}[x^2](x - 4) - x^2\frac{d}{dx}[x - 4]}{(x - 4)^2} = \frac{2x(x - 4) - x^2(1)}{(x - 4)^2} = \frac{2x^2 - 8x - x^2}{(x - 4)^2} = \frac{x^2 - 8x}{(x - 4)^2}$
Stationary Points ( $h'(x) = 0$ ): Set numerator to zero: $x(x - 8) = 0 \implies x = 0 \quad \text{or} \quad x = 8$ Both $x = 0$ and $x = 8$ are in the domain of $h(x)$ .
Singular Points ( $h'(x) \text{ DNE}$ ): The derivative is undefined when $(x - 4)^2 = 0 \implies x = 4$ . However, $x = 4$ is NOT in the domain of $h(x)$ ( $h(4)$ is undefined; the graph has a vertical asymptote at $x = 4$ ).
Conclusion: The critical numbers are strictly $x = 0$ and $x = 8$ . ( $x = 4$ is excluded).

Connections & References

Parent Concept: Definition of the Derivative
Sub-Concepts & Theorems:
Course Sessions:
- Calc 1 Session 13 - Critical Points and Extrema
- Calc 1 Session 14 - Optimizing Areas and Volumes

Tony's Intertwingled Memex • Reference Manual

Critical Numbers and Critical Points

Concept Explanation

1. The Anatomy of a Critical Number

2. Fermat's Theorem and the Converse Trap

3. The Crucial Domain Restriction Rule

Engineering Applications

Example Problems

Example 1: Critical Numbers of a Polynomial Function

Example 2: Critical Numbers with Singular Points (Cusps)

Example 3: Rational Function and the Domain Trap

Connections & References