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

[ausdreamer]

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.

 

[babita]

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

 

[babita]

sry i typed the inequality wrong

 

[tms]

The cases where A = 0 and A = B should be obvious. For the rest, 0 &lt; A &lt; B, think geometrically.