뒤로Second Derivative Test, Concavity, and Inflection Points in Business Calculus
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Q1. For the function :
Background
Topic: Second Derivative Test, Concavity, and Inflection Points
This question tests your understanding of how to use the second derivative to determine where a function is concave upward or downward, and how to find inflection points where the concavity changes.
Key Terms and Formulas
Concave Upward:
Concave Downward:
Inflection Point: A point where changes sign and is continuous.
First Derivative:
Second Derivative:
Step-by-Step Guidance
Find the first derivative of the function .
Find the second derivative by differentiating .
Set and solve for to find possible inflection points.
Test intervals between these -values to determine where (concave upward) and (concave downward).
Check that is continuous at the points where and confirm that the sign of changes to identify inflection points.
Try solving on your own before revealing the answer!
Final Answer:
A) The graph is concave upward on the interval where . For , . So, gives . Thus, concave upward on .
B) The graph is concave downward where , i.e., .
C) The inflection point occurs at . The -coordinate is . So, the inflection point is at .
We found the intervals by analyzing the sign of the second derivative and confirmed the inflection point by checking the sign change and continuity.
Q2. For the function :
Background
Topic: Second Derivative Test, Concavity, and Inflection Points
This question is similar to Q1, focusing on determining concavity and inflection points for a quartic function.
Key Terms and Formulas
Concave Upward:
Concave Downward:
Inflection Point: Where changes sign and is continuous.
Step-by-Step Guidance
Find for .
Find by differentiating .
Solve for to find possible inflection points.
Test intervals between these -values to determine where and .
Check for sign changes and continuity at these points to confirm inflection points.
Try solving on your own before revealing the answer!
Final Answer:
A) The graph is concave upward where . For , . Set to find the intervals.
B) The graph is concave downward where .
C) The inflection points are at and . Plug these into to get the coordinates: and .
We determined the intervals and inflection points by analyzing the sign of the second derivative.
Q3. For the function :
Background
Topic: Second Derivative Test, Concavity, and Inflection Points
This question continues the practice of finding concavity and inflection points for a cubic function.
Key Terms and Formulas
Concave Upward:
Concave Downward:
Inflection Point: Where changes sign and is continuous.
Step-by-Step Guidance
Find for .
Find by differentiating .
Solve for to find possible inflection points.
Test intervals between these -values to determine where and .
Check for sign changes and continuity at these points to confirm inflection points.
Try solving on your own before revealing the answer!
Final Answer:
A) The graph is concave upward where . For , . Set to find the interval.
B) The graph is concave downward where .
C) The inflection point is at . The -coordinate is . So, the inflection point is .
We found the intervals and inflection point by analyzing the sign of the second derivative.
Q4. For the function :
Background
Topic: Second Derivative Test, Concavity, and Inflection Points
This question asks you to analyze a cubic function for concavity and inflection points.
Key Terms and Formulas
Concave Upward:
Concave Downward:
Inflection Point: Where changes sign and is continuous.
Step-by-Step Guidance
Find for .
Find by differentiating .
Solve for to find possible inflection points.
Test intervals between these -values to determine where and .
Check for sign changes and continuity at these points to confirm inflection points.
Try solving on your own before revealing the answer!
Final Answer:
A) The graph is concave upward where . For , . Set to find the interval.
B) The graph is concave downward where .
C) The inflection point is at . The -coordinate is . So, the inflection point is .
We found the intervals and inflection point by analyzing the sign of the second derivative.
Q5. A T-shirt manufacturer: ,
Background
Topic: Diminishing Returns, Concavity, and Inflection Points in Business Applications
This question applies the concept of concavity and inflection points to a real-world business scenario, specifically the point of diminishing returns in production.
Key Terms and Formulas
Rate of Change:
Increasing Rate of Change:
Decreasing Rate of Change:
Point of Diminishing Returns: The value of where and changes from positive to negative.
Step-by-Step Guidance
Find for .
Find by differentiating .
Solve for to find the point of diminishing returns.
Determine the intervals where (rate of change increasing) and (rate of change decreasing).
Check the sign change at the critical value to confirm the point of diminishing returns.
Try solving on your own before revealing the answer!
Final Answer:
A) The rate of change of T-shirt production is increasing where . For , . Set to find the interval.
B) The rate of change is decreasing where .
C) The point of diminishing returns is at new workers.
This is where the rate of change of production is maximized, and adding more workers yields a lower rate of increase in production.
Q6. A baseball cap manufacturer: ,
Background
Topic: Diminishing Returns, Concavity, and Inflection Points in Business Applications
This question is similar to Q5, focusing on the point of diminishing returns for a different production function.
Key Terms and Formulas
Rate of Change:
Increasing Rate of Change:
Decreasing Rate of Change:
Point of Diminishing Returns: The value of where and changes from positive to negative.
Step-by-Step Guidance
Find for .
Find by differentiating .
Solve for to find the point of diminishing returns.
Determine the intervals where (rate of change increasing) and (rate of change decreasing).
Check the sign change at the critical value to confirm the point of diminishing returns.
Try solving on your own before revealing the answer!
Final Answer:
A) The rate of change of baseball cap production is increasing where . For , . Set to find the interval.
B) The rate of change is decreasing where .
C) The point of diminishing returns is at new workers.
This is where the rate of change of production is maximized, and adding more workers yields a lower rate of increase in production.
