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Finding Global Extrema quiz #1

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  • What is a critical point of a function, and how do you find it?
    A critical point is where the derivative of a function is zero or does not exist. To find it, set the derivative equal to zero and solve for x, and also check where the derivative does not exist within the domain.
  • Does every critical point correspond to a local maximum or minimum? Explain.
    No, not every critical point is a local maximum or minimum. Some critical points may not be extrema; they are just points where the derivative is zero or undefined.
  • State the Extreme Value Theorem.
    The Extreme Value Theorem states that if a function is continuous on a closed interval [a, b], then it has both a global maximum and a global minimum on that interval.
  • What are the steps to find the global maximum and minimum of a continuous function on a closed interval?
    First, verify the function is continuous on a closed interval. Next, find all critical points within the interval. Then, evaluate the function at the critical points and at the endpoints. The largest value is the global maximum, and the smallest is the global minimum.
  • Given f(x) = x³ - 12x + 5, how do you find its critical points?
    Find the derivative f'(x) = 3x² - 12, set it equal to zero, and solve for x. This gives x = 2 and x = -2 as critical points.
  • Why is it important to check where the derivative does not exist when finding critical points?
    Because critical points include both where the derivative is zero and where it does not exist, which could indicate a cusp or vertical tangent that may be a local extremum.
  • If a function is not continuous on a closed interval, can you guarantee global extrema? Why or why not?
    No, you cannot guarantee global extrema if the function is not continuous on a closed interval, because the Extreme Value Theorem does not apply.
  • For the function f(x) = 3x² + 1 on the interval [-2, 4], what are the steps to find its global extrema?
    First, find the derivative f'(x) = 6x and set it to zero to get x = 0. Then, evaluate f(x) at x = -2, x = 0, and x = 4. The largest value is the global maximum, and the smallest is the global minimum.
  • What values should you test when searching for global extrema on a closed interval?
    You should test all critical points within the interval and the endpoints of the interval.
  • How do you determine if a function is continuous on a closed interval?
    Check that the function has no breaks, holes, or discontinuities on the interval, and that it is defined at the endpoints.
  • Why do you only consider critical points inside the closed interval when finding global extrema?
    Because only critical points within the interval can affect the global maximum or minimum on that interval; points outside are not relevant.
  • What is the difference between a local extremum and a global extremum?
    A local extremum is the highest or lowest point in a small neighborhood, while a global extremum is the absolute highest or lowest point on the entire interval.
  • If a function has no critical points in a closed interval, how do you find its global extrema?
    Evaluate the function at the endpoints; the larger value is the global maximum and the smaller is the global minimum.
  • Why is it necessary to check both endpoints and critical points when finding global extrema?
    Because the global maximum or minimum can occur at either a critical point or an endpoint of the interval.