# Maximum value(Tricky)

1. Jul 13, 2008

### ritwik06

1. The problem statement, all variables and given/known data
If x belongs set of real numbers, find the maximum value of $$2(a-x)(x+\sqrt{x^{2}+b^{2}})$$

3. The attempt at a solution
All that I could do was to try and diffrentiate the above expression but it yields a polynomial of degree 4. A hint to this question is that it belongs to the Quadratic Equations Chapter. And the answer to th above question is $$a^{2}+b^{2}$$

2. Jul 13, 2008

### tiny-tim

complete the square …

Hi ritwik06!

You have to complete the square (twice).

3. Jul 13, 2008

### ritwik06

Re: complete the square …

I didnt get u tim. Coul u please be more explicit...
thanks

4. Jul 13, 2008

### tiny-tim

Re: complete the square …

Do you know what "complete the square" means?

If you do … then try it!

(If you don't, I'll show you an example.)

5. Jul 13, 2008

### ritwik06

Re: complete the square …

I tried this:
help me!!!
$$b^{2}+2ax+2a(\sqrt{b^{2}+x^{2}})-(x+(\sqrt{b^{2}+x^{2}))^{2}$$
Hey, please help. Dont think that I havnt tried it at all. I have done all that I could. Thanks for the help.

6. Jul 13, 2008

### tiny-tim

Re: complete the square …

ok … two hints:

i] You have two squares to complete, and you need x2 in both of them … so you'll have to split up the -2x2, won't you?

ii] You know what the answer is, so that gives you a pretty good clue as to what might be left over!

(And, as I said, get rid of the 2ax first.)

7. Jul 14, 2008

### ritwik06

Re: complete the square …

I have worked more on this problem but yet no results come my way;
$$((a+\sqrt(x^{2}+b^{2}))^{2})-((x-a)^{2})-((x+\sqrt(x^{2}+b^{2}))^{2})$$

8. Jul 14, 2008

### tiny-tim

Re: complete the square …

ok …

try $$2(a-x)(x+\sqrt{x^{2}+b^{2}}) = a^2\ -\ (a-x)^2\ -\ x^2\ +\ 2(a-x)\sqrt{x^{2}+b^{2}}$$

9. Jul 21, 2008

### Staff: Mentor

It is not a correct answer.

But I must admit I have no idea how to complete these squares. Not that I am surprised, I am mathematically challenged. I have solved it by brute force, which was much easier then expected.

10. Jul 21, 2008

### tiny-tim

completing the square

Hi Borek!

Try adding $b^2\ -\ b^2$ to my last post.

11. Jul 22, 2008

### Staff: Mentor

Arrgh. There is nothing like solving different question that was asked. I was looking for x such that the function given gets maximum value. That's for

$$x = \frac {a^2-b^2} {2a}$$

but

$${a^2+b^2}$$

is a correct max value.

Thanks TT

12. Jul 24, 2008

### ritwik06

Thanks a lot Tim.