Understanding Critical Points and Intervals of Increase/Decrease

The video tutorial explains how to find critical numbers of a function by determining where its first derivative is zero or undefined. It then shows how to use these critical numbers to divide the domain into intervals and test the sign of the derivative to determine where the function is increasing or decreasing. The tutorial also discusses the concept of relative and absolute minima and verifies results using a graph.

What is the main task when given the function f(x) = 4x(x - 3)^(2/3)?

• Determine the critical numbers and intervals of increase or decrease.

• Find the maximum value of the function.

• Calculate the second derivative.

• Solve the function for x.

What is the inner function when applying the chain rule to f(x) = 4x(x - 3)^(2/3)?

• 2/3

• x - 3

• 4x

• x

What is the simplified form of the derivative of f(x) = 4x(x - 3)^(2/3)?

• 4x * (x - 3)^(2/3)

• 8/3 * (x - 3)^(2/3)

• 4/3 * (x - 3)^(1/3)

• 8/3 * (x - 3)^(-1/3)

At what value of x is the derivative of the function undefined?

• x = 0

• x = 4

• x = 3

• x = -3

What is the critical number for the function f(x) = 4x(x - 3)^(2/3)?

• x = -3

• x = 3

• x = 0

• x = 4

What is the sign of the derivative on the interval from negative infinity to 3?

• Positive

• Negative

• Undefined

• Zero

What is the behavior of the function on the interval from 3 to infinity?

• The function is increasing.

• The function is undefined.

• The function is decreasing.

• The function is constant.

What type of minimum does the function have at x = 3?

• No minimum

• Relative minimum

• Both relative and absolute minimum

• Absolute minimum

What is the function value at the minimum point x = 3?

• 4

• 0

• 3

• 1

How can the results be verified?

• By using a calculator.

• By solving the equation.

• By looking at the graph of the function.

• By calculating the second derivative.