Math Help - Differentiation problem (shouldn't be too hard)

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 max 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 max of: 4log(100-X) + 2log(X + Y)

So I need to find X.

Any help would be much appreciated!

Originally Posted by 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 max of: a(1-X)^1/2 + (X)^1/2

where a is a positive constant.
So I need to find out what X equals.

At a local maximum of a differentiable function f we have:

df/dx=0.

In this case:

f(x) = a(1-x)^1/2 + (x)^1/2

so:

df/dx= a(1/2)(1-x)^{-1/2}(-1) + (1/2) x^{-1/2}

so if df/dx=0 we have:

x^{-1/2} = a (1-x)^{-1/2}

squaring:

1/x = a^2/(1-x)

and if x!=0 and x!=1 (you can check neither of these give df/fx=0 so this
is OK), we have:

x=(1-x)/a

so x=1/(1+a)


Now this can be a maximum, a minimum or a point of inflection so we need to
check that this is a maximum using the second derivative test.

d^2f/dx^2 = -a/[4(1-x)^(3/2)] - 1/[4x^{3/2}]

which is negative at x=1/(1+a) (as a>0), hence this is a maximum.

RonL