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) (infinity , + infinity)
(B) (infinity , 3] U [3 , +infinity)
(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
(infinity , 3) U (3 , +infinity)
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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