04 Mar '15 18:162 edits

OK, I have already asked a similar question before but what I want is something a bit different here that I have been frustratingly been completely stuck on but hope someone here can show me the solution:

It concerns the graph for:

f(x) = (x^e)( (1 – x) ^ (c – e) )

where 0 < x < 1 (x is not allowed to be less than 0 nor greater than 1 ) and c and e are both natural numbers (i.e. positive and whole numbers) and e cannot be larger than c (so the (c – e) exponent is never negative ).

I know that the area under the curve (its integral ) of this graph from x=0 to x=1 is exactly:

e!(c – e)! / (c + 1)!

But what I want to know is, what is the real number M that is such that the area under the curve from x=0 to x=M is exactly HALF of that e!(c – e)! / (c + 1)! i.e. e!(c – e)! /

?

In other words, if you cut that area under the curve for f(x) = (x^e)( (1 – x) ^ (c – e) ) between x=0 and x=1 exactly in half with a perfectly vertical line so that there is exactly equal area under the curve either side of that vertical line, exactly where along the x-axis will that vertical line bisect? At x=...?

( Obviously, 0 < M < 1 )

Anyone?

It concerns the graph for:

f(x) = (x^e)( (1 – x) ^ (c – e) )

where 0 < x < 1 (x is not allowed to be less than 0 nor greater than 1 ) and c and e are both natural numbers (i.e. positive and whole numbers) and e cannot be larger than c (so the (c – e) exponent is never negative ).

I know that the area under the curve (its integral ) of this graph from x=0 to x=1 is exactly:

e!(c – e)! / (c + 1)!

But what I want to know is, what is the real number M that is such that the area under the curve from x=0 to x=M is exactly HALF of that e!(c – e)! / (c + 1)! i.e. e!(c – e)! /

**2**(c + 1)!?

In other words, if you cut that area under the curve for f(x) = (x^e)( (1 – x) ^ (c – e) ) between x=0 and x=1 exactly in half with a perfectly vertical line so that there is exactly equal area under the curve either side of that vertical line, exactly where along the x-axis will that vertical line bisect? At x=...?

( Obviously, 0 < M < 1 )

Anyone?