Maximsation Maths Question

[simonStipulates]

Hi everyone.
I have this multiple choice maths problem that essentially asks for the value x where U is maximised in the formula U(x) = 80(logx)^2 - 12(logx)^3. I have posted the full problem to this thread in case I missed anything.
As far as I can tell, the formula has no definitive maximum, though the answer states U as reaching a maximum at 85.
I'm probably misinterpreting the question somehow, but I can't see where. So if anybody could help out, that would be great!

Thanks!

 

[Steven G]

I do not see why they say 85.

 

[HallsofIvy]

The first thing I would do is differentiate U and set the derivative equal to 0: \(\displaystyle U'= 160 log(x)/x- 36(log(x))^2/x= 0\). Multiplying by x, \(\displaystyle 160 log(x)- 36(log(x))^2= 0\) or \(\displaystyle 40 log(x)= 9 (log(x))^2\). x= 1 so that log(x)= 0 is an obvious solution. If x is not 1, we can divide by 9log(x) to get 40/9= log(x). \(\displaystyle x= e^{40/9}\) which is approximately 85.15.

That is not 85 but "close enough for government work"!