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