Finding Extreme Values in Calculus
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Finding Extreme Values in Calculus

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Questions and Answers

What condition must be satisfied for a value f(c) to be classified as a local maximum?

  • f(c) must be the largest value on the entire interval.
  • f(x) must have no larger value for x in the entire domain.
  • f(c) is greater than or equal to f(x) for all x in a neighborhood of c. (correct)
  • f(c) is less than f(x) for all x in a neighborhood of c.
  • How is an absolute minimum defined in relation to local minima?

  • An absolute minimum is defined as the smallest value in the entire range of the function. (correct)
  • An absolute minimum can be larger than some local minimum values.
  • An absolute minimum is always a local minimum.
  • An absolute minimum can occur only at the endpoints of the domain.
  • In the context of extreme values, what distinguishes an absolute maximum from a local maximum?

  • An absolute maximum is the highest point in the entire domain. (correct)
  • An absolute maximum is always smaller than local maxima.
  • An absolute maximum is the highest point among nearby points.
  • An absolute maximum must occur at the endpoints of the interval.
  • Which statement is true regarding local and absolute minima?

    <p>An absolute minimum can exist without local minima.</p> Signup and view all the answers

    What property does a local minimum at point c indicate about the function values nearby?

    <p>f(c) is less than or equal to f(x) for all x in a neighborhood of c.</p> Signup and view all the answers

    When identifying extreme values, what is a critical point for a continuous function?

    <p>A point where the first derivative is zero or undefined.</p> Signup and view all the answers

    Why might a function have an absolute maximum at a point that is not a local maximum?

    <p>It can be influenced by values at endpoints of the domain.</p> Signup and view all the answers

    Which of the following statements about extreme values is false?

    <p>An absolute maximum is less than or equal to any local maximum.</p> Signup and view all the answers

    What is the maximum value of the function f(x) = ln(x/(1 + x^2))?

    <p>-0.69</p> Signup and view all the answers

    At which points does the function f(x) have extreme values due to its even nature?

    <p>x = 1 and x = -1</p> Signup and view all the answers

    What is the value of x where the function f(x) = ln(x/(1 + x^2)) has a minimum value?

    <p>0</p> Signup and view all the answers

    Which of the following is a critical point of the function f(x) = ln(x/(1 + x^2))?

    <p>x = 1</p> Signup and view all the answers

    What does the derivative f'(x) = (1 - x^2)/(x(1 + x^2)) indicate about the function's behavior?

    <p>It changes sign at the critical points</p> Signup and view all the answers

    Which statement about the extreme values of the function derived graphically is true?

    <p>The maximum value occurs at multiple points.</p> Signup and view all the answers

    How is it confirmed analytically that f(x) = ln(x/(1 + x^2)) has extreme values?

    <p>By setting the derivative equal to zero.</p> Signup and view all the answers

    What does the function f(x) = ln(x/(1 + x^2)) lead to in terms of its domain?

    <p>All nonzero real numbers</p> Signup and view all the answers

    What is the nature of the critical point at x = 0 for the function f(x)?

    <p>It is an absolute minimum.</p> Signup and view all the answers

    What must be true for the Extreme Value Theorem to guarantee extreme points?

    <p>The function must be continuous on a closed interval.</p> Signup and view all the answers

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

    <p>The function has no maxima, local or absolute.</p> Signup and view all the answers

    How does the function f(x) behave as x moves away from the critical point?

    <p>The values of f increase indefinitely.</p> Signup and view all the answers

    What is the derivative f¿(x) at the critical point x = 0?

    <p>0</p> Signup and view all the answers

    Why can the function f(x) be confirmed to have a minimum at x = 0?

    <p>It is the only critical point.</p> Signup and view all the answers

    What happens to the function f(x) when x is in the interval (-2, 2)?

    <p>The function only attains a minimum value.</p> Signup and view all the answers

    Which phrase best describes the behavior of f(x) near its critical point?

    <p>It rises as you move away from the critical point.</p> Signup and view all the answers

    Study Notes

    Local and Absolute Extreme Values

    • Absolute maximum is the greatest value of a function on an interval and is also a local maximum
    • Local maximum is the greatest value of a function in a neighborhood
    • Absolute minimum is the smallest value of a function on an interval and is also a local minimum
    • Local minimum is the smallest value of a function in a neighborhood

    Finding Extreme Values

    • Extreme values are the maximum and minimum values of a function within the domain of the function
    • Local extreme values can be found using a function's derivative
    • To find local extreme values find the critical points of a function, where the derivative is 0 or undefined, or where it does not exist
    • Absolute extreme values can be found using a function's derivative or by graphing it
    • When considering absolute extreme values the closed interval of a function's domain must be considered
    • If the interval is open, the function may have local extreme values but not absolute extreme values
    • If the interval is closed, the function will absolutely have absolute extreme values

    Finding Minimum And Maximum for f(x) = ln(1/1+x^2)

    • The function is even and has a domain of all non-zero real numbers
    • The function has maximum values at points close to x = 1 and x = -1
    • The values of the function at x = 1 and x = -1 are both about -0.69
    • The derivative of the function is (1-x^2)/(x(1+x^2))
    • The derivative is not defined at x = 0
    • The critical points of the function are at x = 1 and x = -1 where the derivative is equal to 0
    • The function does not have an absolute minimum value
    • The function has a relative minimum at x = 0

    Finding Minimum And Maximum for f(x) = 1/(24-x^2)

    • The domain of the function is -2 ≤ x ≤ 2
    • The only critical point within the domain is x = 0, where the derivative is equal to 0
    • The value of the function at x = 0 is 1/24
    • The function has a minimum value at x = 0 and the minimum is absolute
    • The function does not have any maxima, either local or absolute
    • This is because the function is defined on an open interval and not a closed interval, and a closed interval is needed for the function to have absolute extreme values

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    Description

    This quiz covers the concepts of local and absolute extreme values in calculus. It focuses on how to identify maximum and minimum values of functions using derivatives and critical points. Understand how to distinguish between local and absolute extrema within different intervals.

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