Unit 5: Analytical Applications of Differentiation

The First Derivative Test

Finding critical points and classifying local extrema using the first derivative.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Where a Graph Turns, According to Its Slope

A function's graph turns from rising to falling, or falling to rising, exactly where its instantaneous rate of change crosses zero — or fails to exist. Before reading on: if you found every x where f'(x) = 0 or is undefined, how would you tell which of those points are peaks, which are valleys, and which are neither?

Definition — Critical Point and the First Derivative Test

A critical point of f is a value c in f's domain where f'(c) = 0 or f'(c) is undefined. The first derivative test classifies each one: if f' changes from positive to negative at c, f has a local maximum there; if f' changes from negative to positive, a local minimum; if f' doesn't change sign, c is neither.

Worked Example — Classifying Critical Points with a Sign Analysis

Find and classify the critical points of f(x) = x³ − 3x² − 9x + 5. f'(x) = 3x² − 6x − 9 = 3(x² − 2x − 3) = 3(x − 3)(x + 1). Critical points: x = −1 and x = 3 (both in the domain, all real numbers). Test a point in each of the three resulting intervals: f'(−2) = 3(−5)(−1) = 15 > 0 (increasing on (−∞, −1)); f'(0) = 3(−3)(1) = −9 < 0 (decreasing on (−1, 3)); f'(4) = 3(1)(5) = 15 > 0 (increasing on (3, ∞)). f' changes from + to − at x = −1: a local maximum, f(−1) = −1 − 3 + 9 + 5 = 10. f' changes from − to + at x = 3: a local minimum, f(3) = 27 − 27 − 27 + 5 = −22.

Worked Example — A Critical Point That Isn't an Extremum

Find the critical points of f(x) = x³. f'(x) = 3x², which equals 0 only at x = 0 — a critical point. But 3x² ≥ 0 for every x, so f' never actually changes sign around x = 0: it's positive on both sides. Since f' doesn't switch from positive to negative or negative to positive, x = 0 is neither a local maximum nor a local minimum — a reminder that f'(c) = 0 alone never guarantees an extremum.

Derivative Explorer

f(x) =

x^3 - 3*x

Point of tangency

x = a1

Show tangent line

Trace mode

f(x), with tangent line at x = a

Show f''(x)Show f'''(x)

Derivative graph — linked to the graph above; panning or zooming either moves both

What's happening at x = a

At x = 1, f(x) ≈ -2 — the point (1, -2).
The derivative there is f'(1) ≈ 0: the slope of the tangent line, and the instantaneous rate of change of f at this exact point.
That slope is positive, so f is increasing at this point.
The second derivative is positive here, so f is concave up (curving upward) near this point.
Near x = 1, the tangent line y ≈ -2 + 0·(x − 1) is f's best straight-line approximation — the core idea a derivative captures.

Numerical derivative comparison

Forward, backward, and central difference estimates of f'(a) as the step size h shrinks — all three should converge toward the same value.
h	Forward	Backward	Central
1	4	-2	1
0.1	0.31	-0.29	0.01
0.01	0.0301	-0.0299	0.0001
0.001	0.003	-0.003	0

Analysis of f(x)

y-intercept: (0, 0)
x-intercepts: (-1.73, 0), (0, 0), (1.73, 0)
Extrema: local max at (-1, 2); local min at (1, -2)
Inflection points: (0, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-970, 970]

Analysis of f'(x) — its roots are f's critical points

y-intercept: (0, -3)
x-intercepts: (-1, 0), (1, 0)
Extrema: local min at (0, -3)
Inflection points: none found in view
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-3, 297]

Tip

A sign analysis only needs one test point per interval between consecutive critical points — f' can't change sign without passing through zero or an undefined point, so its sign stays constant across an entire interval between critical points.

Common Mistakes

Assuming every solution of f'(c) = 0 is automatically a local extremum.
f'(c) = 0 only makes c a critical point, a candidate — f(x) = x³ at x = 0 shows a critical point where the derivative doesn't actually change sign, so it's neither a max nor a min.
Forgetting to check where f'(x) is undefined, and only solving f'(x) = 0.
A critical point can also occur where f' fails to exist (a corner or vertical tangent in f, for instance) — both cases have to be checked, not just the zeros of f'.

Key Takeaways

A critical point occurs where f'(c) = 0 or f'(c) is undefined, and is only a candidate for a local extremum, not a guarantee.
The first derivative test classifies a critical point by whether f' changes from positive to negative (local max), negative to positive (local min), or doesn't change sign (neither).
A sign analysis needs only one test value per interval between consecutive critical points, since f' can't change sign without crossing zero or an undefined point.

Summary

The first derivative test reads a graph's shape from f'. The next lesson reads a second layer of shape — concavity — from f'', and offers a faster alternative for classifying extrema.

Track your progress

Back to Unit 5: Analytical Applications of Differentiation The Second Derivative Test & Concavity→