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Math Help - a,b,c,d are real ,prove this

  1. #1
    Like a stone-audioslave ADARSH's Avatar
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    a,b,c,d are real ,prove this


    Given a,b,c,d as 4 real numbers

    Avoiding method of contradiction prove

     <br />
a-b^2 , b - c^2 , c-d^2 , d- a^2 <br />
    cant all be greater
    than \frac{1}{4}?
    Thanks to the helper
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  2. #2
    Super Member flyingsquirrel's Avatar
    Joined
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    Hi,
    Quote Originally Posted by ADARSH View Post
    Given a,b,c,d as 4 real numbers

    Avoiding method of contradiction prove

     <br />
a-b^2 , b - c^2 , c-d^2 , d- a^2 <br />
    cant all be greater
    than \frac{1}{4}?
    Let f(a,b,c,d)=(a-b^2)+(b-c^2)+(c-d^2)+(d-a^2).

    <br />
\begin{aligned}f(a,b,c,d)&=(a-b^2)+(b-c^2)+(c-d^2)+(d-a^2)\\<br />
 &=1{\color{blue}-a^2+a-\frac14}{\color{green}-b^2+b-\frac14}{\color{red}-c^2+c-\frac14}-d^2+d-\frac14\\<br />
&=1\underbrace{{\color{blue}-\left(a-\frac12\right)^2}{\color{green}-\left(b-\frac12\right)^2}{\color{red}-\left(c-\frac12\right)^2}-\left(d-\frac12\right)^2}_{\leq 0}\\<br />
f(a,b,c,d)&\leq 1\\<br />
\end{aligned}

    Can you take it from here ?
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