__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
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