Inequality of arithmetic and geometric means

[Numeriprimi]

Hey! I have this: 2(√(1-a^2 ))+ 2a
How to determine the maximum value of this?

I think good for this is Inequality of arithmetic and geometric means, but I don't know how use this, because I don't calculate with this yet.

So, have you got any ideas?



Poor Czech Numeriprimi... If you don't understand my primitive English, write, I will try to write this better ;-)

 

[symbolipoint]

Hey! I have this: 2(√(1-a^2 ))+ 2a
How to determine the maximum value of this?

Could be a Calculus derivative problem. Does this f(a)=2(√(1-a^2 ))+ 2a have a maximum or minimum?

 

[HallsofIvy]

Carrying that farther f(a)= 2(1- a^2)^{1/2}+ 2a has derivative f'(a)= -2(1- a^2)^{-1/2}+ 2. Any maximum (or minimum) must occur where that derivative is 0 or does not exist. It does not exist when a= 1 or -1 because in that case we have a 0 in the denominator. It is 0 when \sqrt{1- a^2}= 1 or a= 0.

Check the values of 2\sqrt{1- a^2}+ 2a at x= -1, 0, and 1 to see which is the maximum value.

Another way to look at this is to see that if y= 2\sqrt{1- a^2}+ 2a then y- 2a= 2\sqrt{1- a^2} and so y^2- 4ay+ 4a^2= 4- 4a^2 or 8a^2- 4ay+ y^2= 4 which is the graph of an ellipse with major and minor axes rotated from the coordinate axes.

 

[Numeriprimi]

Please, derivate no... I still have a few years time to learn this.
I need to use Inequality of arithmetic and geometric means, bud how?

 

[AlephZero]

Your expression has two parts, $2\sqrt{1-a^2}$ amd $2a$.

If you are supposed to use arithmetic and geometric means, you need to find some quantities that give those arithmetic and geometric means.

For example if the quantities are $x$ and $y$ and the arithmetic mean is $2a$, you have the equation $(x+y)/2 = 2a$

And you have another equation for the geometric mean ...

 

[Numeriprimi]

Yes, i have this: n=4
And what now?