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MATH / Jonathon is given the following problem: Let a and b be positive real numbers such that a+b=1. Find the minimum value of (a+(1)/(a))²+(b+(1)/(b))².Jonathon reasons as follows: By A…

Problem

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problem

Jonathon is given the following problem: Let aa and bb be positive real numbers such that a+b=1.a + b = 1. Find the minimum value of (a+1a)2+(b+1b)2.\left( a + \frac{1}{a} \right)^2 + \left( b + \frac{1}{b} \right)^2.Jonathon reasons as follows: By AM-GM, a+1a≥2,a + \frac{1}{a} \ge 2,and b+1b≥2,b + \frac{1}{b} \ge 2,so (a+1a)2+(b+1b)2≥22+22=8.\left( a + \frac{1}{a} \right)^2 + \left( b + \frac{1}{b} \right)^2 \ge 2^2 + 2^2 = 8.He concludes that the minimum value is 8, but this answer is incorrect. What is the minimum value? As a bonus, determine where Jonathon went wrong.
Plain-text mathematical notation (without MathML)
Jonathon is given the following problem: Let a and b be positive real numbers such that a+b=1.  Find the minimum value of
(a+(1)/(a))²+(b+(1)/(b))².Jonathon reasons as follows: By AM-GM,
a+(1)/(a)≥2,and
b+(1)/(b)≥2,so
(a+(1)/(a))²+(b+(1)/(b))²≥2²+2²=8.He concludes that the minimum value is 8, but this answer is incorrect.  What is the minimum value?

As a bonus, determine where Jonathon went wrong.
Original LaTeX notation
Jonathon is given the following problem: Let $a$ and $b$ be positive real numbers such that $a + b = 1.$  Find the minimum value of
\[\left( a + \frac{1}{a} \right)^2 + \left( b + \frac{1}{b} \right)^2.\]Jonathon reasons as follows: By AM-GM,
\[a + \frac{1}{a} \ge 2,\]and
\[b + \frac{1}{b} \ge 2,\]so
\[\left( a + \frac{1}{a} \right)^2 + \left( b + \frac{1}{b} \right)^2 \ge 2^2 + 2^2 = 8.\]He concludes that the minimum value is 8, but this answer is incorrect.  What is the minimum value?

As a bonus, determine where Jonathon went wrong.

level

Level 4

type

Intermediate Algebra

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Source and history

task_id
64b5d63b-b976-51e7-96d4-f2f5f9695f8e
task_key
intermediate~5falgebra--test--64b5d63b-b976-51e7-96d4-f2f5f9695f8e
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import