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Calculus I · Unit 2B · lesson

The First Derivative Test

Concept

Learning objectives

Use changes in the sign of f′f' to classify local extrema.

Classify Local Extrema by Sign Changes

Explanation

Before the formulas

Graph analysis in The First Derivative Test is a coherent reconstruction problem. Domain, intercepts, limits, derivative signs, critical points, concavity, and asymptotes constrain the same curve. Build the picture in layers rather than trying to sketch from the original formula at once.

Keep a feature table. Each row should state the calculation, the interval or point, and the graphical consequence. This makes the final sketch a summary of evidence instead of an artistic guess.

Critical numbers divide the domain into intervals. Testing the sign of f' on each interval determines whether the original function rises or falls and classifies sign changes.
Read this graph as text

A derivative sign chart turns algebra into a graph story. Critical numbers divide the domain into intervals. Testing the sign of f' on each interval determines whether the original function rises or falls and classifies sign changes. A positive derivative means the original function increases; a negative derivative means it decreases. At x=-2 , the sign changes from positive to negative, so f has a local maximum. At x=1 , the sign changes from negative to positive, so f has a local minimum.

Every relationship in a derivative sign chart turns algebra into a graph story is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.

Why it matters: The chart should bridge symbolic factor signs and graph behavior. It is a reusable component for increasing/decreasing intervals, the first derivative test, and optimization verification.

Visual study

Critical numbers divide the domain into intervals. Testing the sign of f' on each interval determines whether the original function rises or falls and classifies sign changes.

Explanation

A turning point is detected by a sign change

At a local maximum, the function changes from increasing to decreasing, so f′f' changes from positive to negative. At a local minimum, f′f' changes from negative to positive. If the sign does not change, the critical point is not a local extremum.

This test describes behavior on both sides and works even when f′f' is undefined at the critical point, provided the original function is defined there.

The first derivative test classifies a critical point by observing what the function does on either side. A change from increasing to decreasing produces a local maximum; decreasing to increasing produces a local minimum; no sign change produces neither.

This test handles cases where f"f" is zero or undefined, so it is often more robust than the second derivative test. It also explains the result through actual motion of the graph rather than a memorized sign table.

Theorem

First Derivative Test

At a critical number cc:

• if f′f' changes from positive to negative, ff has a local maximum; • if f′f' changes from negative to positive, ff has a local minimum; • if f′f' does not change sign, there is no local extremum.

For the previous cubic, f′f' changes +→−+\to- at −1-1, so ff has a local maximum there. It changes −→+-\to+ at 33, so ff has a local minimum there.

Guided walkthrough

A critical point that is not an extremum

Classify the critical point of f(x)=x3f(x)=x^3 at 00.

Answer reveal

Worked solution

Write a real attempt before opening the supplied answer.

Common mistake

Solving f′(x)=0f'(x)=0 finds candidates. It does not classify them. A zero derivative may be a maximum, minimum, flat inflection point, or part of a constant interval.

Modeling lab

Peak concentration after a dose

Suppose

C(t)=20te−0.5tC(t)=20te^{-0.5t}

models drug concentration for t≥0t\ge0. Then

C′(t)=20e−0.5t(1−0.5t).C'(t)=20e^{-0.5t}(1-0.5t).

The exponential factor is positive, so the derivative changes sign when 1−0.5t=01-0.5t=0, at t=2t=2. Concentration increases before 22 hours and decreases after, so the first derivative test identifies a peak at t=2t=2.

After the explanation

Use the section idea

Reading lens

Turn derivative signs and theorem hypotheses into a defensible account of extrema, monotonicity, concavity, and global shape.

Mental model

Critical numbers divide the domain into testable intervals; endpoints and discontinuities keep local evidence from becoming an unjustified global claim.

Decision

List the domain and candidates, test derivative signs, compare endpoint values, and verify each theorem's hypotheses explicitly.

Common trap

Calling every point with f-prime zero an extremum or every point with f-double-prime zero an inflection point.

Check yourself

Can every turn, bend, endpoint result, and asymptote in your sketch be traced to algebraic evidence?

Interactive checkapp-drug-peak-01

For C(t)=20te−0.5tC(t)=20te^{-0.5t}, at what time is concentration maximal?

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Show hint

Factor C′(t)C'(t) and locate its sign change.

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