Questions on the critical numbers of functions are presented below[cite: 1]. These questions are designed to help you gain a deep understanding of critical numbers in calculus. Each solution is hidden inside a collapsible dropdown so you can practice independently before reviewing the steps.
A critical number \(c\) of a function \(f\) is a number in the domain of \(f\) such that:
Answer: (C)
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.
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.
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.
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.
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 \(\frac{1}{k}\).
More references on calculus: questions with answers[cite: 1] and tutorials and problems[cite: 1].