Q2. Does f(x) achieve a relative maximum when x = -4?

Background

Topic: Relative Extrema from Graphs

This question tests your ability to identify relative maxima by analyzing the graph of a function. A relative maximum occurs at a point where the function changes from increasing to decreasing.

Key Terms:

• Relative Maximum: A point where the function value is higher than all nearby points.

• Critical Point: A point where the derivative is zero or undefined, and where extrema may occur.

Step-by-Step Guidance

1. Locate x = -4 on the x-axis of the graph.

2. Observe the behavior of the function near x = -4. Is the function increasing before x = -4 and decreasing after?

3. Check if the point at x = -4 is a peak (highest point in its immediate neighborhood).

4. Consider whether the function is defined at x = -4 (open or closed circle) and if the slope changes from positive to negative.

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Q3. Is the absolute maximum value of f(x) equal to 4?

Background

Topic: Absolute Maximum from Graphs

This question asks you to determine the highest value that the function attains on its domain, based on the graph.

Key Terms:

• Absolute Maximum: The largest value of f(x) over the entire domain.

Step-by-Step Guidance

1. Scan the graph for the highest point reached by the function.

2. Identify the y-value at this highest point.

3. Check if this value is 4 and if it is attained at a specific x-value.

4. Verify that there are no higher points elsewhere on the graph.

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Q4. Is the absolute minimum value of f(x) equal to ?

Background

Topic: Absolute Minimum from Graphs

This question tests your understanding of how to determine if a function is unbounded below (i.e., decreases without bound) by analyzing the graph.

Key Terms:

• Absolute Minimum: The lowest value of f(x) over the entire domain.

• Unbounded Below: The function decreases without limit as x increases or decreases.

Step-by-Step Guidance

1. Look at the behavior of the graph as x approaches positive or negative infinity.

2. Check if the function continues to decrease without reaching a lowest point.

3. Determine if there is any horizontal asymptote or if the graph goes downward indefinitely.

4. Assess whether the minimum value is finite or .

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