Critical Numbers and Derivatives

The video tutorial explains how to determine the critical numbers of a function by finding where its first derivative is zero or undefined. It begins by discussing the domain of the function, which is crucial for identifying valid critical numbers. The tutorial then demonstrates how to find the derivative using the product rule and simplifies the expression. Finally, it solves the equation to find the critical numbers and analyzes the graph to identify relative maxima.

What are critical numbers in the context of a function?

• Values of x where the function is undefined

• Values of x where the first derivative is zero or undefined

• Values of x where the function has a maximum

• Values of x where the function has a minimum

What is the domain of the function f(x) = x^(-3) * ln(x)?

• x < 0

• All real numbers

• x > 0

• x ≠ 0

Why can't x equal zero in the function f(x) = ln(x) / x^3?

• Because ln(x) is zero at x = 0

• Because ln(x) is undefined at x = 0

• Because x^3 is undefined at x = 0

• Because it would result in division by zero

Which rule is used to find the derivative of f(x) = x^(-3) * ln(x)?

• Product rule

• Quotient rule

• Chain rule

• Power rule

What is the derivative of ln(x)?

• e^x

• ln(x)

• x

• 1/x

What is the simplified form of the derivative of f(x) = x^(-3) * ln(x)?

• 1/x^3 - 3ln(x)/x^3

• x^4 - 3ln(x)x^4

• 1/x^4 - 3ln(x)/x^4

• x^3 - 3ln(x)x^3

Tags

CCSS.HSF-IF.C.7D

What is the critical number found for the function f(x) = x^(-3) * ln(x)?

• x = 1/e

• x = e^3

• x = e^(1/3)

• x = 3e

What does the critical number x = e^(1/3) indicate about the function?

• The function is undefined

• The function has a relative maximum

• The function has a relative minimum

• The function is constant

What is the approximate value of the critical number x = e^(1/3)?

• 3.142

• 0.693

• 2.718

• 1.396

What does the graph of the function indicate at x = e^(1/3)?

• A low point

• A high point

• A point of inflection

• A vertical asymptote