Understanding Function Behavior

The video tutorial explains how to find the open intervals where a function is increasing or decreasing and identify any local extrema. It begins by discussing the domain of the function, which excludes zero. The tutorial then covers finding critical numbers by rewriting the function and calculating its first derivative. The domain is divided into intervals to test the sign of the derivative, determining where the function increases or decreases. A calculator is used to evaluate the derivative at test values, and the results are verified graphically. Finally, the video calculates and confirms the values of local maxima and minima.

What is the domain of the given function?

• All real numbers

• All real numbers except zero

• Only negative numbers

• Only positive numbers

What is the first derivative of the function 4x + 3/x?

• 4x - 3

• 4 - 3x^2

• 4 + 3/x^2

• 4 - 3/x^2

What are the critical numbers of the function?

• x = ±1

• x = 0

• x = ±2

• x = ±√3/2

How many intervals is the domain divided into using the critical numbers?

• Two

• Three

• Four

• Five

Which test value is used for the interval from negative infinity to negative √3/2?

• x = -0.5

• x = -1

• x = 0.5

• x = 1

What is the sign of the first derivative at x = -1?

• Zero

• Negative

• Undefined

• Positive

What is the behavior of the function over the interval from -√3/2 to 0?

• Increasing

• Decreasing

• Undefined

• Constant

What is the local maximum value of the function?

• Approximately 0

• Approximately 3.4641

• Approximately -6.9282

• Approximately 6.9282

At which point does the function have a local minimum?

• x = 1

• x = -√3/2

• x = √3/2

• x = 0

What is the approximate value of the local minimum?

• 3.4641

• 0

• -6.9282

• 6.9282