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When $a\neq b$ $a,b\in\mathbb{R}$ find the minimum of $\frac{a^2+b^2}{2}-\frac{|a-b|}{2}\sqrt{1+(a+b)^2}$

I think that the answer is $-\frac{1}{4}$ but not sure. Please tell me a solution.

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  • $\begingroup$ How did you arrive at your answer? $\endgroup$ Apr 30, 2022 at 22:57
  • $\begingroup$ Have you tried differentiating the function? $\endgroup$ Apr 30, 2022 at 22:58
  • $\begingroup$ I thought that the answer will be when a=-b and then it could be written as $a^2-a$ from this the minimum should be $-\frac{1}{4}$ when $a=\frac{1}{2}$ $\endgroup$
    – user998872
    Apr 30, 2022 at 23:28
  • $\begingroup$ I could not prove that it should be $a=-b$ $\endgroup$
    – user998872
    Apr 30, 2022 at 23:29
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    $\begingroup$ Using $(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)$, we have $$\frac{a^2+b^2}{2}-\frac{|a-b|}{2}\sqrt{1+(a+b)^2} = \left(\frac{|a - b|}{2} - \frac 12\sqrt{1 + (a + b)^2}\right)^2 - \frac14.$$ $\endgroup$
    – River Li
    May 1, 2022 at 6:05

1 Answer 1

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Since swapping $a$ and $b$ does not change the value of the expression, let's restrict ourselves only to solutions where $a\geq b$. This allows us to rewrite $|a-b|$ as $a-b$, which makes our expression much easier to deal with.

Let $c=\frac{a+b}{2}$ and $d=\frac{a-b}{2}$. Then $c$ can be any real number, and $d$ can be any non-negative real. Written in terms of $c$ and $d$, our expression becomes:

$c^2 + d^2 - d\sqrt{4c^2+1}$

Differentiating with respect to $d$ gives $2d - \sqrt{4c^2+1}$. This is zero when $d = \sqrt{c^2 + 1/4}$ which means this value of $d$ minimises the expression.

If you plug this value of $d$ into the expression, the $c$s cancel out and the expression reaches its minimum value of $-1/4$ as you suggested.

If you have a particular value for $a$ and you want a corressponding value of $b$ that achieves this minimum, remember that $a=c+d$ and $d = \sqrt{c^2 + 1/4}$. Solving this gives $c=\frac{a}{2}-\frac{1}{8a}$ and $d=\frac{a}{2}+\frac{1}{8a}$. We get $b=c-d =-\frac{1}{4a}$. This only works if this value of $b$ is less than or equal to $a$, which we see is only when $a>0$.

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