Differentiation: Find max of a(1-X)^1/2+X^1/2, where a is...

[roblomas]

Hi, I have two equations that i need to find the max of...apparently this is finding the derivative of the function..but I'm struggling..

1. Find the maximum of a(1-X)^1/2 + (X)^1/2, where a is a positive constant.

So I need to find out what X equals.

2. Find the maximum of 4log(100-X) + 2log(X + Y)

So I need to find X.

Any help would be much appreciated!

 

[soroban]

Re: Differentiation: Find max of a(1-X)^1/2+X^1/2, where a i

Hello, roblomas!

I think you need to review your Algebra: handing fractions . . .

1. Find the maximum of \(\displaystyle f(x)\:=\:a(1\,-\,x)^{\frac{1}{2}}\,+\,x^{\frac{1}{2}}\), where \(\displaystyle a\) is a positive constant.

Differentiate and equate to zero: \(\displaystyle \L\:f'(x)\:=\:a\cdot\frac{1}{2}(1\,-\,x)^{-\frac{1}{2}}\cdot(-1)\,+\,\frac{1}{2}x^{-\frac{1}{2}} \:=\:0\)

We have: \(\displaystyle \L\:\frac{-a}{2\sqrt{1\,-\,x}}\,+\,\frac{1}{2\sqrt{x}} \:=\:0\;\;\Rightarrow\;\;\frac{a}{\sqrt{1\,-\,x}}\:=\:\frac{1}{\sqrt{x}}\;\;\Rightarrow\;\;a\sqrt{x}\:=\:\sqrt{1\,-\,x}\)


Square both sides: \(\displaystyle \L\,a^2x\:=\:1\,-\,x\;\;\Rightarrow\;\;a^2x\,+\,x\:=\:1\;\;\Rightarrow\;\;(a^2\,+\,1)x\:=\:1\)

Therefore: \(\displaystyle \L\:\fbox{x\:=\:\frac{1}{a^2\,+\,1}}\)

2. Find the maximum of: \(\displaystyle \,f(x)\:=\:4\cdot\ln(100\,-\,x)\,+\,2\cdot\ln(x\,+\,y)\)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .↑?

Is that really a "y" . . . or a typo?

 

[roblomas]

Re: Differentiation: Find max of a(1-X)^1/2+X^1/2, where a i

2. Find the maximum of: \(\displaystyle \,f(x)\:=\:4\cdot\ln(100\,-\,x)\,+\,2\cdot\ln(x\,+\,y)\)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .↑?

Is that really a "y" . . . or a typo?

Hi Soroban, thanks so much for the solution to the first one!

Yeh the second one is more confusing. Basically I need to find either T1 or T2 and i was told to use differentiation to do this (I think because it needs to be the max)..


f = 4logX + 2logT, where X=100-T1 and T = T1+T2

that's how I got:

f = 4log(100-T1) + 2log(T1+T2)


If you have any methods for finding either T1 or T2 using differentiation that would be awesome. Thanks again

Rob.

 

[soroban]

Re: Differentiation: Find max of a(1-X)^1/2+X^1/2, where a i

Hello again, Rob!

I'm still puzzled . . .

\(\displaystyle f \:=\: 4\cdot\ln x\,+\,2\cdot\ln T\), where \(\displaystyle x\:=\:100\,-\,T_1\) and \(\displaystyle T\:=\:T_1\,+\,T_2\)

that's how I got: \(\displaystyle \:f \:=\: 4\cdot\ln(100\,-\,T_1)\,+\,2\cdot\ln(T_1\,+\,T_2)\)

If you have any methods for finding either \(\displaystyle T_1\) or \(\displaystyle T_2\) . . .

So we are dealing with a function \(\displaystyle f\) with two variables ?

Are you familiar with partial derivatives?
. . If not, there is some information missing from the problem.


Those subscripts are annoying.
. . Let \(\displaystyle u \,=\,T_1,\;v\,=\,T_2\)

Then we have: \(\displaystyle \:f(u,v) \:=\:4\cdot\ln(100\,-\,u)\,+\,2\cdot\ln(u\,+\,v)\)

But the max/min procedure fails us.

. . \(\displaystyle \begin{array}{ccccc}\frac{\partial f}{\partial u} & \,=\, & \frac{-4}{100\,-\,u}\,+\,\frac{2}{u\,+\,v} & \,=\, & 0 \\ \\ \\

\frac{\partial f}{\partial v} & \,=\, & \frac{2}{u\,+\,v} & \,=\, & 0\end{array}\)

The second equation has no solutions, you see . . .