Q1. Find the absolute maximum and absolute minimum of on .

Background

Topic: Absolute Extrema on a Closed Interval

This question tests your ability to find the absolute maximum and minimum values of a continuous function on a closed interval. This is a key concept in Business Calculus, especially for optimization problems.

Key Terms and Formulas

• Absolute Maximum: The largest value of on the interval.

• Absolute Minimum: The smallest value of on the interval.

• Critical Point: A point in the domain where or does not exist.

Steps to find absolute extrema on :

• Find and solve to get critical points inside the interval.

• Evaluate at all critical points and at the endpoints and .

• The largest value is the absolute maximum; the smallest is the absolute minimum.

Step-by-Step Guidance

1. Find the derivative: .

2. Set and solve for to find any critical points within .

3. List all candidates for extrema: the critical points from step 2 and the endpoints and .

4. Evaluate at each candidate point to determine which values are the largest and smallest.

Try solving on your own before revealing the answer!

Q2. Find the absolute maximum and absolute minimum of on (A) and (B) .

Background

Topic: Absolute Extrema for Quadratic Functions on Closed Intervals

This question asks you to find the highest and lowest values of a quadratic function on two different intervals. This is a common type of optimization problem in Business Calculus.

Key Terms and Formulas

• Quadratic Function:

• Critical Point: For a quadratic, occurs at

• Check endpoints and any critical points within the interval.

Step-by-Step Guidance

1. Find the derivative: .

2. Set and solve for to find the critical point.

3. For each interval, check if the critical point lies within the interval.

4. Evaluate at the endpoints and at the critical point (if it is in the interval).

Try solving on your own before revealing the answer!

Q3. Find the absolute maximum and absolute minimum of on .

Background

Topic: Absolute Extrema for Cubic Functions on Closed Intervals

This question involves finding the highest and lowest values of a cubic function on a closed interval, which is important for understanding optimization in calculus.

Key Terms and Formulas

• Cubic Function:

• Critical Points: Solve for in the interval.

• Check endpoints and all critical points within the interval.

Step-by-Step Guidance

1. Find the derivative: .

2. Set and solve for to find all critical points in .

3. List all candidates: the critical points and the endpoints and .

4. Evaluate at each candidate to compare their values.

Try solving on your own before revealing the answer!

Q4. Find the absolute maximum and absolute minimum of on .

Background

Topic: Absolute Extrema for Cubic Functions on Closed Intervals

This question asks you to find the absolute maximum and minimum values of a cubic function on a closed interval, a common application in business optimization problems.

Key Terms and Formulas

• Cubic Function:

• Critical Points: Solve for in the interval.

• Check endpoints and all critical points within the interval.

Step-by-Step Guidance

1. Find the derivative: .

2. Set and solve for to find all critical points in .

3. List all candidates: the critical points and the endpoints and .

4. Evaluate at each candidate to compare their values.

Try solving on your own before revealing the answer!