Absolute Minimum and Maximum Word Problem

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SUMMARY

The discussion focuses on finding a number within the closed interval [1/2, 3/2] that minimizes and maximizes the function S = x + 1/x. The minimum value occurs at x = 1, yielding S = 2, while the maximum value occurs at x = 1/2, resulting in S = 2. The derivative dS/dx = 1 - 1/x^2 is set to zero to locate local extrema, and endpoints must also be evaluated to confirm these results.

PREREQUISITES
  • Understanding of calculus, specifically derivatives and critical points
  • Familiarity with closed intervals in real analysis
  • Knowledge of reciprocal functions
  • Ability to evaluate functions at endpoints
NEXT STEPS
  • Study the concept of local extrema in calculus
  • Learn about evaluating functions on closed intervals
  • Explore optimization techniques in calculus
  • Review the properties of reciprocal functions
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Students studying calculus, particularly those focusing on optimization problems, as well as educators seeking to explain concepts related to derivatives and extrema.

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Homework Statement


Find a number in the closed interval [1/2, 3/2] such that the sum of the number and its reciprocal is
(a)as small as possible
(b) as large as possible

I am given the answer in the back of the book
The answer to a is 1
The answer to be is 1/2




Homework Equations


Here is my equation

S= x + 1/x

dS/dx = 1-1/x^2

The Attempt at a Solution


I was wondering if i should set the derivative equal to zero and find the zeroes for the derivatives?
 
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Absolutely, you should do that. That will give the locations of the local extrema. And don't forget to check the endpoints as well.
 
Last edited:

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