Understanding Decreasing Intervals of a Function

The video tutorial explores how to determine the intervals where a function is decreasing using derivatives. It begins with defining the function and explaining that a function decreases when its derivative is negative. The derivative is calculated using the power rule, and the expression is simplified by factoring. The tutorial then analyzes the conditions under which the product of terms is negative, leading to the identification of intervals where the function decreases. The conclusion is supported by graphing the function to visually confirm the analysis.

What is the function f(x) that we are analyzing?

• f(x) = x^6 - 3x^5

• f(x) = x^5 - 3x^6

• f(x) = x^6 + 3x^5

• f(x) = x^5 + 3x^6

When is a function considered to be decreasing?

• When its derivative is zero

• When its derivative is positive

• When its derivative is negative

• When its derivative is undefined

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

• f'(x) = 5x^6 - 15x^5

• f'(x) = 6x^5 - 15x^4

• f'(x) = 5x^5 - 15x^4

• f'(x) = 6x^6 - 15x^5

What is the simplified form of the derivative f'(x) after factoring?

• 3x^4(5x - 2)

• 3x^5(2x - 5)

• 3x^5(5x - 2)

• 3x^4(2x - 5)

Why is the condition 3x^4 < 0 impossible?

• Because x^4 is always negative

• Because x^4 is always positive

• Because x^4 can be zero

• Because x^4 is undefined

For which values of x is 3x^4 > 0 true?

• For all x

• For x > 0

• For x < 0

• For x ≠ 0

What is the interval where the function is decreasing?

• x > 5/2

• x ≠ 5/2

• x < 5/2

• x = 5/2

What happens to the function at x = 0?

• It decreases

• It increases

• It remains constant

• It is undefined

Why is the function still decreasing at x = 0?

• Because the derivative is positive

• Because the derivative is negative

• Because the derivative is zero but the function decreases on both sides

• Because the function is undefined at x = 0

What tool was used to graph the function and verify the intervals?

• Graphing calculator

• Desmos

• Mathematica

• Excel