Critical Numbers of Functions

Questions on the critical numbers of functions are presented below. These questions are designed to help you gain a deep understanding of critical numbers in calculus. Answers are also provided.

Questions and Solutions

Question 1

A critical number \(c\) of a function \(f\) is a number in the domain of \(f\) such that:

Answer: (C)

Question 2

True or False: The function \(f(x) = |x|\) has no critical points.
Answer: False. The derivative is \[ f'(x) = \frac{x}{|x|} \] which is undefined at \(x = 0\). Therefore, \(x = 0\) is a critical point.

Question 3

True or False: If \(c\) is a critical number, then \(f(c)\) is either a local maximum or minimum.
Answer: False. For example, \(f(x) = x^3\) has a critical number at \(x = 0\), but \(f(0)\) is neither a local maximum nor minimum.

Question 4

True or False: If \(c\) is not a critical number, then \(f(c)\) is neither a local minimum nor maximum.
Answer: True. This is the contrapositive of Fermat's theorem: if \(f(c)\) is a local maximum or minimum, then \(c\) must be a critical number.

Question 5

The values of parameter \(a\) for which the function \[ f(x) = x^3 + ax^2 + 3x \] has two distinct critical numbers are in the interval:

Answer: D Derivative: \[ f'(x) = 3x^2 + 2ax + 3 \] Solve \(f'(x) = 0\) to find critical numbers. The discriminant is \[ D = (2a)^2 - 4(3)(3) = 4a^2 - 36 \] \(D > 0\) for \(a \in (-\infty, -3) \cup (3, +\infty)\), giving two distinct solutions.

Question 6

If \(f(x)\) has one critical point at \(x = c\), then:

Answer: (E) Shifting the graph horizontally shifts the critical point. Horizontal compression by \(k\) scales the critical point by \(\frac{1}{k}\).

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