Test on Extrema of Functions

Instructions

Please read each question carefully and select the best answer. The questions will cover concepts related to finding extrema (maximum and minimum values) of functions, including local and global extrema, first and second derivative tests, and applications.

Multiple Choice Questions

Question 1: Definition of Extrema

What is the definition of a local maximum of a function ( f(x) )?

A) The value of ( f(x) ) at a critical point where ( f(x) ) is higher than the values of ( f(x) ) in its neighborhood.
B) The value of ( f(x) ) at the endpoint of a function's domain.
C) The highest value of ( f(x) ) in the entire domain of the function.
D) The average value of ( f(x) ) over its entire domain.

Answer: A) The value of ( f(x) ) at a critical point where ( f(x) ) is higher than the values of ( f(x) ) in its neighborhood.

Question 2: Identifying Critical Points

Which of the following statements is true about critical points of a function ( f(x) )?

A) Critical points occur where ( f'(x) = 0 ) or ( f(x) ) is not differentiable.
B) Critical points always correspond to local maxima.
C) Every critical point must be a global maximum or minimum.
D) Critical points can only be determined using the second derivative test.

Answer: A) Critical points occur where ( f'(x) = 0 ) or ( f(x) ) is not differentiable.

Question 3: First Derivative Test

Using the first derivative test, how can you determine whether a critical point is a local maximum, local minimum, or neither?

A) Check the sign of ( f'(x) ) before and after the critical point.
B) Calculate ( f''(x) ) at the critical point.
C) Evaluate the function at the critical point compared to nearby points only.
D) None of the above.

Answer: A) Check the sign of ( f'(x) ) before and after the critical point.

Question 4: Second Derivative Test

If ( f''(x) > 0 ) at a critical point ( x = c ), what can you conclude?

A) The function has a local maximum at ( x = c ).
B) The function has a local minimum at ( x = c ).
C) The function is neither maximum nor minimum at ( x = c ).
D) The critical point cannot be assessed.

Answer: B) The function has a local minimum at ( x = c ).

Question 5: Global Extrema

Consider the function ( f(x) = -x^2 + 4x ). What are the global maximum and minimum values of this function over the interval ([0, 4])?

A) Maximum: 8, Minimum: 0
B) Maximum: 4, Minimum: 0
C) Maximum: 4, Minimum: -8
D) Maximum: 8, Minimum: -4

Answer: B) Maximum: 4, Minimum: 0

Short Answer Questions

Question 6: Finding Extrema

Using the function ( f(x) = x^3 - 3x^2 + 4 ), find the critical points and determine if they are maxima or minima.

Answer:

First, find ( f'(x) = 3x^2 - 6x ).
Set ( f'(x) = 0 ) to get critical points: ( 3x^2 - 6x = 0 ) gives ( x(x - 2) = 0 ), so ( x = 0 ) and ( x = 2 ).
Using the second derivative test, find ( f''(x) = 6x - 6 ):
- At ( x = 0 ): ( f''(0) = -6 < 0 ) (local maximum).
- At ( x = 2 ): ( f''(2) = 6 > 0 ) (local minimum).

Thus, ( x = 0 ) is a local maximum and ( x = 2 ) is a local minimum.

Question 7: Applications of Extrema

Describe a real-world situation where finding the local or global extrema of a function would be essential.

Answer: One application might be in the field of economics, specifically in profit maximization. A company may use a profit function ( P(x) ) that represents profit based on the quantity of goods produced ( x ). To identify how many units to produce to maximize profit, the company must find the local maximum of the profit function. This involves taking the derivative of the profit function, finding critical points, and determining global or local extrema to make informed production and pricing decisions.

Conclusion

By completing this test, you should have a clearer understanding of the concepts related to extrema of functions. Review the correct answers and explanations to strengthen your knowledge in this area!

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