Why isn't the answer to TMUA-P2-ERR2-009 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P2-ERR2-009 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P2-ERR2-009?

No — calculators are banned in the TMUA. TMUA-P2-ERR2-009 is set for a no-calculator target of about 270 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P2-ERR2-009?

TMUA-P2-ERR2-009 is Paper P2 · Common fallacies (specification code ERR2), level 3. Drill more of the same topic from this page.

TMUA-P2-ERR2-009 — Paper P2 · Common fallacies · Level 3

P2ERR2Level 3~270s target

Spec: ERR2 · No calculator · ~270s

A sixth-former optimises a profit function

P(x)

and argues as follows: Fermat's theorem says that if

P

has a local maximum at an interior point

c

then

P'(c) = 0

; the student finds

P'(c) = 0

c = 40

and immediately announces a maximum profit there, without checking endpoints, sign changes, or second-order behaviour. The teacher provides the graph showing

c = 40

is actually a point of inflection with the true maximum at the domain endpoint. Which fallacy did the student commit?

Options

No fallacy: a stationary point is always a maximum.
Affirming the consequent: $P'(c) = 0$ is necessary but not sufficient for a local maximum, so the converse inference “stationary therefore maximum” is invalid.
Denying the antecedent of a true theorem.
A valid use of the contrapositive.
The student divided by zero.

Letters only — the answer key is hidden. Work it out, then check the solution below.

Hint (check trope)

No written hint yet — check the trope: necessary-vs-sufficient-swap, converse-vs-contrapositive. Name the trap first, work the stem by hand, then open the solution below.

Show solution

Fermat gives max

\Rightarrow

stationary (necessary); the student ran it backwards as stationary

\Rightarrow

max (sufficient) — the converse error in calculus clothing. Trap: fluent “set derivative to zero” routines hide that endpoints, saddles and minima share

P'(c) = 0

; classification needs further tests.

Keep practising

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Options

No fallacy: a stationary point is always a maximum.

Affirming the consequent:

P'(c) = 0

is necessary but not sufficient for a local maximum, so the converse inference “stationary therefore maximum” is invalid.

Denying the antecedent of a true theorem.

A valid use of the contrapositive.

The student divided by zero.

Letters only — the answer key is hidden. Work it out, then check the solution below.

Keep practising

Same topic · same level

P2ERR2Level 3

TMUA-P2-ERR2-010

A philosophy essay argues: “Everyone has a biological mother. Therefore there is someone who is everyone's biological mo…

Common fallacies

Open TMUA-P2-ERR2-010 →

Same trap: converse-vs-contrapositive

P2ARG1Level 1

TMUA-P2-ARG1-001

Let P and Q be mathematical statements. The converse of the statement “If P then Q ” is which of the following?

Statements and connectives

Open TMUA-P2-ARG1-001 →

TMUA-P2-ERR2-009 — Paper P2 · Common fallacies · Level 3

Options

Keep practising

Master Common fallacies

TMUA-P2-ERR2-009 — Paper P2 · Common fallacies · Level 3

Options

Keep practising

Master Common fallacies