Critical Numbers of Functions

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 :

Question 2

True or False. Function f defined by f(x) = | x | has no critical points.
Answer :
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 :
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 :
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 + a x 2 + 3x

has two distinct critical numbers are in the interval
(A) (-∞ , + ∞)
(B) (-∞ , -3] U [3 , +∞)
(C) (0 , + infinty)
(D) None of the above
The derivative of f is given by
f(x) = 3 x 2 + 2 a x + 3

The critical numbers may be found by solving
f '(x)= 3 x 2 + 2 a x + 3 = 0
The discriminant D of the above quadratic equation is given by
D = (2 a) 2 - 4(3)(3) = 4 a 2 - 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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