# 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.

 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) = 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 + 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 Answer: D. 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. More references on calculus questions with answers and tutorials and problems .