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Critical Numbers of Functions

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) = 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) = x³ 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) = x³ + a x² + 3x
has two distinct critical numbers are in the interval
(A) (-∞ , + ∞)
(B) (-∞ , -3] U [3 , +∞)
(C) (0 , + infinty)
(D) None of the above
Answer:
D.
The derivative of f is given by
f(x) = 3 x² + 2 a x + 3
The critical numbers may be found by solving
f '(x)= 3 x² + 2 a x + 3 = 0
The discriminant D of the above quadratic equation is given by
D = (2 a)² - 4(3)(3) = 4 a² - 36
D is positive and the quadratic equation has two distinct solutions for a in the interval
(-∞ , -3) U (3 , +∞)

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(k x) has a critical point at x = 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. But if you shift the graph of a function you also shift its critical point(s). f(k x) 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.

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