How can I prove the inequality A(B-A) <= (B/2)^2 for 0 <= A <= B?

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Homework Help Overview

The problem involves proving the inequality A(B-A) ≤ (B/2)² under the conditions 0 ≤ A ≤ B. This falls within the subject area of inequalities and algebraic manipulation.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss rearranging terms and consider the implications of the conditions on A and B. Some suggest a geometric interpretation for the case where 0 < A < B.

Discussion Status

The discussion is ongoing, with participants exploring different approaches and questioning the necessity of the given conditions. Some guidance has been offered regarding algebraic manipulation and geometric reasoning.

Contextual Notes

There is a noted confusion regarding the essential nature of the conditions 0 ≤ A ≤ B, as well as a correction of a typographical error in the inequality. The participants are navigating these constraints while attempting to formulate a proof.

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Homework Statement



If 0 <= A <= B, prove that: A(B-A) <= (B/2)^2

Homework Equations



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The Attempt at a Solution



I've been blindly rearranging the terms trying to see a way to prove this but due to my complete lack of experience in proofs, I'm hoping someone here can give a little push in a helpful direction.
 
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try opening the brackets and taking all the terms to one side. it'll become square of a number.
but i dnt understand how 0>=a>=b are essential conditions for this. square of any real no would always be positive
 
sry i typed the inequality wrong
 
The cases where A = 0 and A = B should be obvious. For the rest, 0 &lt; A &lt; B, think geometrically.
 

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