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 x =
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