Home / AP Calculus AB 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema – MCQs

AP Calculus AB 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema - MCQs - Exam Style Questions

AP Calculus AB-MCQs and Answers - All Topics

No-Calc Question

Let \(g\) be the function defined by \(g(x)=x^{4}+4x^{3}\). How many relative extrema does \(g\) have?

(A) Zero
(B) One
(C) Two
(D) Three

▶️ Answer/Explanation

Differentiate: \(g'(x)=4x^{3}+12x^{2}=4x^{2}(x+3)\)
Critical points: \(x=0,\,-3\)
Test intervals show a sign change only at \(x=-3\)
Relative extrema: 1
✅ Answer: (B) One

No-Calc Question

If \(f'(x)=(x-2)(x-3)^{2}(x-4)^{3}\), then \(f\) has which of the following relative extrema?

I. A relative maximum at \(x=2\)
II. A relative minimum at \(x=3\)
III. A relative maximum at \(x=4\)

(A) I only
(B) III only
(C) I and III only
(D) II and III only
(E) I, II, and III

▶️ Answer/Explanation

Test sign changes of \(f'(x)\) across the critical points \(x=2,3,4\):

Interval\(x-2\)\((x-3)^2\)\((x-4)^3\)Sign of \(f'(x)\)
\((-\infty,2)\)−+−+
\((2,3)\)++−−
\((3,4)\)++−−
\((4,\infty)\)++++

At \(x=2\): sign changes \(+\) to \(−\) ⇒ relative maximum.
At \(x=3\): even multiplicity, no sign change ⇒ no extremum.
At \(x=4\): sign changes \(−\) to \(+\) ⇒ relative minimum (not a maximum).
Answer: (A) I only.

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