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 SolutionsQuestion 1A 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 2True or False. Function f defined by f(x) =  x  has no critical points.Answer : False. The derivative f '(x) is given by 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 3True 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 4True 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 5The values of parameter a for which function f defined byhas 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 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 6If 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 .
