hi,i want to ask 1 question the question is like this if a and b are positive numbers,find the maximum value of f(x)=x^a(1-x)^b where x is more and equal to 0 and x is less and equal to 1.

Question

hi,i want to ask 1 question the question is like this if a and b are positive numbers,find the maximum value of f(x)=x^a(1-x)^b where x is more  and equal to 0 and x is less and equal to 1.

zepdrix · Answer

Hi,
$\Large\bf \color{#008353}{\text{Welcome to OpenStudy! :)}}$

zepdrix · Answer

Mmm so what are we doing?
Using basic calc methods to figure this out?

zepdrix · Answer

$$\Large\rm f(x)=x^a(1-x)^b,\qquad 0\le x\le 1$$

zepdrix · Answer

$$\Large\rm a>0, \qquad b>0$$

anonymous · Answer

it is ok if i just assume any positive number for a and b?

zepdrix · Answer

Well your final answer will probably involve a and b, so you don't want to plug in values to try and simplify the problem, I think that might end up causing some trouble.
If that's what you were asking..

zepdrix · Answer

Applying product rule to take our derivative:
$$\Large\rm f'(x)=a x^{a-1}(1-x)^b-b x^a(1-x)^{b-1}$$That step make sense?

anonymous · Answer

i have try product rule and i've got like this
 ax^(a−1)(1−x)^b=x^ab(1−x)^(b−1)

ax^(a−1)x^(−a)(1−x)^b=b(1−x)^(b−1)

ax^−1(1−x)^b=b(1−x)^(b−1)

a=xb(1−x)^(b−1)(1−x)^−b

a=xb(1−x)^−1

a/b=x/1−x

a−ax=bx

a=bx+ax

a=x(b+a)

x=a/(b+a)