Questions on the critical numbers of functions are presented. The present questions have been designed to help you gain deep understanding of the concept of a critical number of a function as defined in calculus. Answers to these questions are also presented.

Questions and Solutions

Question 1

A critical number \( c \) of a function \( f \) is a number in the domain of \( f \) such that
(A) \( f'(c) = 0 \)
(B) \( f'(c) \) is undefined
(C) (A) or (B) above
(D) None of the above
Answer :
(C).

Question 2

True or False. Function \( f \) defined by \( f(x) = | x | \) has no critical points.
Answer :
False.
The derivative \( f'(x) \) is given by
\( f'(x) = \dfrac{x}{|x|} \)
\( f'(x) \) is undefined at \( x = 0 \) and therefore \( x = 0 \) is a critical point of function \( f \) given above. (see question 1 above)

Question 3

True or False. If \( c \) is a critical number then \( f(c) \) is either a local maximum or a local minimum.
Answer :
False.
\( f(x) = x^3 \) has a critical number at \( x = 0 \) yet \( f(0) \) is neither a local maximum nor a local minimum.

Question 4

True or False. If \( c \) is not a critical number then \( f(c) \) is neither a local minimum nor a local maximum.
Answer :
True.
This is the contrapositive of Fermat's theorem: If \( f(c) \) is a local maximum or local minimum then \( c \) must be a critical number of \( f \).

Question 5

The values of parameter \( a \) for which function \( f \) defined by
\( f(x) = x^3 + ax^2 + 3x \)
has two distinct critical numbers are in the interval
(A) \((- \infty , + \infty)\)
(B) \((- \infty , -3] \cup [3 , +\infty)\)
(C) \((0 , + \infty)\)
(D) None of the above
Answer:
D.
The derivative of \( f \) is given by
\( f(x) = 3x^2 + 2ax + 3 \)
The critical numbers may be found by solving
\( f'(x) = 3x^2 + 2ax + 3 = 0 \)
The discriminant \( D \) of the above quadratic equation is given by
\( D = (2a)^2 - 4(3)(3) = 4a^2 - 36 \)
\( D \) is positive in the interval \((- \infty , -3) \cup (3 , +\infty)\) and therefore the quadratic equation has two distinct solutions for \( a \) in the interval \((- \infty , -3) \cup (3 , +\infty)\).

Question 6

If \( f(x) \) has one critical point at \( x = c \), then
(A) function \( f(x - a) \) has one critical point at \( x = c + a \)
(B) function \( -f(x) \) has a critical point at \( x = -c \)
(C) \( f(kx) \) has a critical point at \( x = \dfrac{c}{k} \)
(D) None of the above
(E) (A) and (C) only
Answer :
(E). The graph of \( f(x - a) \) is the graph of \( f(x) \) shifted \( a \) units to the right. If you shift the graph of a function you also shift its critical point(s). (A) is true.
\( f(kx) \) is about the horizontal compression of the graph of a function. If the graph of a function is compressed horizontally then its critical point(s) is also compressed horizontally. (C) is true.