Understanding Extrema in Functions

The video tutorial explains how to determine the relative and absolute extrema of the function f(x) = -3 log x over the domain 0 < x ≤ 10. It highlights the presence of a vertical asymptote at x = 0 and the decreasing nature of the function. The tutorial identifies the absolute minimum at the endpoint (10, -3) but clarifies that it does not qualify as a relative minimum due to endpoint constraints. The video concludes with a summary of the findings, emphasizing the absence of a relative minimum.

What is the function given in the problem?

• f(x) = 3x

• f(x) = -3x

• f(x) = -3 log x

• f(x) = 3 log x

What is the domain of the function f(x) = -3 log x?

• x >= 0 and x <= 10

• x > 0 and x < 10

• x > 0 and x <= 10

• x >= 0 and x < 10

What happens to the function values as x approaches zero?

• They decrease without bound

• They remain constant

• They increase without bound

• They oscillate

Why is there no absolute maximum for the function?

• The function has a horizontal asymptote

• The function is decreasing and has a vertical asymptote

• The function is constant

• The function is increasing

What is the absolute minimum value of the function?

• 0

• -3

• 10

• 3

At which x-value does the absolute minimum occur?

• x = 0

• x = 10

• x = 15

• x = 5

According to the definition, when does a function have an absolute minimum?

• When f(c) is not defined

• When f(c) is equal to all other values

• When f(c) is less than or equal to all other values

• When f(c) is greater than all other values

Why does the point (10, -3) not qualify as a relative minimum?

• It is not in the domain

• It is the highest point

• It is an endpoint and cannot be approached from both sides

• It is not the lowest point

What is required for a point to be considered a relative minimum?

• It must be outside the domain

• It must be in an open interval

• It must be an endpoint

• It must be the highest point

What does DNE stand for in the context of this problem?

• Does Not Evaluate

• Does Not Exist

• Does Not Equal

• Does Not End