Maximum value of 2(a-x)(x+√(x²+b²)) for real x

[ritwik06]

Homework Statement

If x belongs set of real numbers, find the maximum value of $$2(a-x)(x+\sqrt{x^{2}+b^{2}})$$

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 $$a^{2}+b^{2}$$
Please help me. I can't make head or tail out of it.

 

[tiny-tim]

complete the square …

If x belongs set of real numbers, find the maximum value of $$2(a-x)(x+\sqrt{x^{2}+b^{2}})$$
…
A hint to this question is that it belongs to the Quadratic Equations Chapter. And the answer to th above question is $$a^{2}+b^{2}$$

Hi ritwik06!

You have to complete the square (twice).

Hint: start with the easy bit … the 2ax.

 

[ritwik06]

Hi ritwik06!

You have to complete the square (twice).

Hint: start with the easy bit … the 2ax.

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

 

[tiny-tim]

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

Do you know what "complete the square" means?

If you do … then try it!

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

 

[ritwik06]

Do you know what "complete the square" means?

If you do … then try it!

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

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.

 

[tiny-tim]

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.

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.)

 

[ritwik06]

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.)

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

 

[tiny-tim]

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

ok …

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

 

[Borek]

And the answer to th above question is $$a^{2}+b^{2}$$

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.

 

[tiny-tim]

completing the square

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.

Hi Borek!

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

 

[Borek]

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

 

[ritwik06]

Thanks a lot Tim.
I had got my answer the day you had posted your second last reply.
Thank you very very much. I am sorry for xpressing my gratitude a bit late. Thanks again.