# 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
(C).

### Question 2

True or False. Function $$f$$ defined by $$f(x) = | x |$$ has no critical points.
False.
The derivative $$f'(x)$$ is given by
$$f'(x) = \dfrac{x}{|x|}$$
$$f'(x)$$ 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.
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.
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
$$f(x) = x^3 + ax^2 + 3x$$
has two distinct critical numbers are in the interval
(A) $$(- \infty , + \infty)$$
(B) $$(- \infty , -3] \cup [3 , +\infty)$$
(C) $$(0 , + \infty)$$
(D) None of the above
D.
The derivative of $$f$$ is given by
$$f(x) = 3x^2 + 2ax + 3$$
The critical numbers may be found by solving
$$f'(x) = 3x^2 + 2ax + 3 = 0$$
The discriminant $$D$$ of the above quadratic equation is given by
$$D = (2a)^2 - 4(3)(3) = 4a^2 - 36$$
$$D$$ is positive in the interval $$(- \infty , -3) \cup (3 , +\infty)$$ and therefore the quadratic equation has two distinct solutions for $$a$$ in the interval $$(- \infty , -3) \cup (3 , +\infty)$$.

### 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) $$f(kx)$$ has a critical point at $$x = \dfrac{c}{k}$$
(D) None of the above
(E) (A) and (C) only
The graph of $$f(x - a)$$ is the graph of $$f(x)$$ shifted $$a$$ units to the right. If you shift the graph of a function you also shift its critical point(s). (A) is true.
$$f(kx)$$ 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. (C) is true.