First Derivative Test

When analysing the shape of a curve, one of the most important questions we can ask is: where does the function reach its highest or lowest values? These special points — called local maxima and local minima — collectively known as local extrema, reveal the turning behaviour of a function. The First Derivative Test gives us a systematic, algebraic method to locate and classify these extrema by examining how the sign of the derivative f'(x) changes around a critical point. Recall that a critical point occurs where f'(x) = 0 or where f'(x) is undefined (and f(x) itself is still defined). At these x-values the tangent line is horizontal or does not exist, making them candidates for extrema. However, not every critical point is an extremum — some are merely flat points where the function continues in the same direction. The First Derivative Test distinguishes between these cases by checking whether the derivative changes sign as x passes through the critical point. The power of the First Derivative Test lies in its directness: you only need to evaluate the sign of f'(x) on either side of each critical point, not compute a second derivative or perform additional algebra. This makes it an essential tool in curve sketching, optimisation problems, and real-world modelling wherever you need to identify peaks, valleys, and plateau points of a function.

Key takeaways
  • Find All Critical Points
  • Create a Sign Chart for f'(x)
  • Apply the Classification Rules
  • State the Extremum Values

Find All Critical Points

Compute f'(x) by differentiating f(x). Set f'(x) = 0 and solve for x. Also note any x-values in the domain where f'(x) is undefined. These x-values are your critical points c₁, c₂, c₃, …

Create a Sign Chart for f'(x)

The critical points divide the real number line (or the domain) into open intervals. Choose one test value inside each interval and evaluate the sign of f'(x) at that test value — you do not need the exact numerical value, only whether f'(x) is positive (+) or negative (−). Record these signs in a table.

Apply the Classification Rules

For a critical point x = c: (a) If f'(x) changes from positive to negative as x increases through c, then f has a LOCAL MAXIMUM at x = c — the function was rising, reached a peak, then started falling. (b) If f'(x) changes from negative to positive as x increases through c, then f has a LOCAL MINIMUM at x = c — the function was falling, reached a valley, then started rising. (c) If f'(x) does not change sign (stays + or stays −), then x = c is neither a local max nor a local min; it is sometimes called a saddle point or inflection point with a horizontal tangent.

State the Extremum Values

Once the type of each extremum is determined, compute the actual y-value by substituting back: the local maximum value is f(c) when f' changes from + to −, and the local minimum value is f(c) when f' changes from − to +. Always report both the x-location and the function value when stating an extremum.

The Key Equation

If f′(c)=0 and {f′(x)>0 for x<cf′(x)<0 for x>c⇒local max at x=c\text{If } f'(c) = 0 \text{ and } \begin{cases} f'(x) > 0 \text{ for } x < c \\ f'(x) < 0 \text{ for } x > c \end{cases} \Rightarrow \text{local max at } x = c
  • f'(x) — First derivative of f (units of f per unit of x): Measures the instantaneous rate of change (slope of the tangent) of f at each point x.
  • c — Critical point (x-units): An x-value where f'(c) = 0 or f'(c) is undefined; a candidate location for a local extremum.
  • f(c) — Extremum value (y-units): The actual output value of the function at the critical point; the height of the peak or depth of the valley.

Worked Example

Problem. Find and classify all local extrema of f(x) = x³ − 3x² − 9x + 5 using the First Derivative Test.

Local max: f(−1)=10;Local min: f(3)=−22\text{Local max: } f(-1) = 10; \quad \text{Local min: } f(3) = -22

Answer. Local maximum of 10 at x = −1; local minimum of −22 at x = 3.

Figure
Diagram
First Derivative Sign Chart: f(x) = x³ − 3x² − 9x + 5x = −1x = 3xf ′ > 0f ′ < 0f ′ > 0(rising)(falling)(rising)+ → − : LOCAL MAXf(−1) = 10− → + : LOCAL MINf(3) = −22MaxMinSchematic curve shape (not to scale)
Origins · 1 minPierre de Fermat and how this was discovered
PD
Pierre de Fermat
1607–1665 · French

Fermat developed an early method for finding maxima and minima of algebraic expressions — well before the formal invention of calculus. His technique of 'adequality' (setting f(x+e) ≈ f(x) and simplifying) anticipated what we now call setting the derivative equal to zero, directly foreshadowing the modern concept of critical points used in the First Derivative Test. Fermat was a lawyer by profession and pursued mathematics entirely as a hobby, yet he made foundational contributions to number theory, probability, and the early ideas of calculus — often scribbling results in the margins of books rather than publishing them formally.

  • Critical Points — The First Derivative Test begins by locating critical points — the x-values where f'(x) = 0 or is undefined — so a solid understanding of critical points is essential prerequisite knowledge.
  • Second Derivative Test — An alternative method for classifying extrema that uses the concavity (sign of f''(c)) instead of sign changes in f', allowing quick classification when the second derivative is easy to compute.
  • Increasing and Decreasing Functions — The sign of f'(x) directly determines whether f is increasing or decreasing on an interval, which is the foundational idea that makes the First Derivative Test work.
  • Concavity and Inflection Points — After identifying local extrema with the First Derivative Test, concavity analysis completes the curve sketch by revealing where the function bends upward or downward.