00:01
To find the maximum value of the function, the first thing you have to do is to take the derivative of this function.
00:08
So by product rule, we have f prime of x that's equal to x raised to a times, you have derivative of 1 minus x raised to b is b times 1 minus x, times negative 1 plus we have 1 minus x raised to b times the derivative of x raised to a that's a x raised to a minus 1 and then we will factor out 1 minus x raised to b minus 1 times x raised to a minus 1 this will be multiplied by minus b times x plus we have a times 1 minus x.
00:52
Simplifying that we have 1 minus x raised to b minus 1 times x raised to a minus 1, this times a minus x times a plus b.
01:05
After taking the derivative we want to set the derivative to 0 to find the critical points of the function.
01:13
So then you will have 1 minus x raised to b minus 1 times x raised to a minus 1 times a minus x times a plus b this is equal to 0 gives us 1 minus x raised to b minus 1 equals 0 you also have x raised to a minus 1 equals 0 and then lastly we have a minus x times a plus b equals 0 and then from here we will solve for x solving for x we have for the first equation we get x equal to 1.
01:49
For the second equation, we get x equal to 0.
01:53
And for the last equation, we get x as equal to a over a plus b.
02:01
Now since the values of a and b are positive numbers, we then assume that a over a plus b is in between 0 and 1...
