Home / AP Calculus AB: 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema – Exam Style questions with Answer- MCQ

AP Calculus AB: 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema – Exam Style questions with Answer- MCQ

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

Ans:B

 First find the critical values where the derivative is 0 or undefined. A relative extrema occurs at a critical value if the derivative changes sign there \(g'(x)=4x^{3}+12x^{2}=x^{2}(4x+12)\)

\(x^{2}(4x+12)=0\) at x = 0 or x = −3

There are two zeros, but g'( x) changes sign for x = −3 only. The sign of g'( x) at x = 0 does not change because of the x2 factor

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 maximum at x=3

III. a relative maximum at x=4

A I

B III

C I and III

D II and III only

E I,II and III

▶️Answer/Explanation

Ans:A

Question

The function defined by  \(f(x)=x^{3}-3x^{2}\) for all real numbers x has a relative maximum at =

A -2

B 0

C 1

D 2

E 4

▶️Answer/Explanation

Ans:B

Question

At what values of x does \(f(x)=3x^{5}-5x^{3}+15\) have a relative maximum?
A -1 only
B 0 only
C 1 only
D -1 and 1 only
E -1, 0 and 1

▶️Answer/Explanation

Ans:A

Scroll to Top