Finding Absolute Extrema in Calculus

Finding Absolute Extrema

• Step 1: Find the critical points of the function within the given interval by differentiating the function and setting the derivative equal to zero.
• Step 2: Evaluate the function at the critical points and endpoints of the interval
• Step 3: The largest value obtained in Step 2 is the absolute maximum, and the smallest value is the absolute minimum.

Problem 1: Finding the Absolute Extrema of g(t) = 8t - t4

• The derivative of g(t) is g'(t) = 8 - 4t3.
• Setting g'(t) = 0, we get t = ³√2.
• Evaluating g(t) at the critical point (³√2) and endpoints (-2, 1), we get:
  • g(-2) = -16
  • g(³√2) ≈ 10.59
  • g(1) = 7
• Therefore, the absolute maximum of g(t) is approximately 10.59 at t = ³√2, and the absolute minimum is -16 at t = -2.

Problem 2: Finding the Absolute Extrema of g(x) = x²/³

• The derivative of g(x) is g'(x) = (2/3)x^(-1/3).
• While the derivative is undefined at x = 0, this point is within the interval.
• Evaluating g(x) at the critical point (0) and endpoints (-2, 3), we get:
  • g(-2) = 2^(2/3) ≈ 1.59
  • g(0) = 0
  • g(3) = 3^(2/3) ≈ 2.08
• Therefore, the absolute maximum of g(x) is approximately 2.08 at x = 3, and the absolute minimum is 0 at x = 0.

This quiz focuses on finding absolute extrema of functions in calculus. You will learn how to identify critical points, evaluate endpoints, and determine maximum and minimum values of given functions. Test your understanding with examples and practice problems.