Understanding Decreasing Intervals of a Function

What is the main objective when analyzing the function f(x) in this tutorial?

• To calculate the integral of f(x).

• To find the maximum value of f(x).

• To determine the intervals where f(x) is increasing.

• To find the intervals where f(x) is decreasing.

Which mathematical rule is used to find the derivative of f(x)?

• Quotient rule

• Chain rule

• Power rule

• Product rule

What is the derivative of the function f(x) = x^6 - 3x^5?

• 6x^5 - 15x^4

• 5x^6 - 3x^5

• x^6 - 3x^5

• 6x^5 + 15x^4

What is the first step in determining when the derivative is less than zero?

• Factoring the derivative

• Solving for x

• Graphing the function

• Integrating the derivative

What factor is common in the terms of the derivative 6x^5 - 15x^4?

• 6x^4

• 3x^4

• x^5

• 15x^3

Why is it impossible for x^4 to be less than zero?

• Because x^4 is always negative.

• Because x^4 is always zero.

• Because x^4 is undefined.

• Because x^4 is always positive or zero for real numbers.

What condition must be true for 3x^4 to be greater than zero?

• x must be equal to zero.

• x must not be equal to zero.

• x must be less than zero.

• x must be greater than zero.

What inequality must be satisfied for 2x - 5 to be negative?

• x > 5/2

• x < 5/2

• x = 5/2

• x > 0

What are the intervals where the function f(x) is decreasing?

• x = 0 or x = 5/2

• x > 0 or x > 5/2

• x < 0 or 0 < x < 5/2

• x < 0 or x > 5/2

Why is the first derivative important in determining the behavior of f(x)?

• It helps find the maximum value of f(x).

• It indicates where f(x) is constant.

• It shows where f(x) is decreasing.

• It determines the integral of f(x).