does anyone know how to do this
find x and z that maximizes ln(4-x) + ln(4+z) + ln(z)
subject to
2 lnx + ln(4-z) + ln(4-z) = 2ln(32/7)
anyhelp is much appreciated
the values of x and z that maximise ln(4-x) + ln(4+z) + ln(z), also
maximise:
(16-x^2)z ...(1)
as x>0, the constraint may be rewritten:
x^2(4-z)^=(32/7)^2,
or:
x=32/(7(4-z)) ...(2)
So substitute (2) into (1) and find the z that maximises this now
unconstrained objective, find the corresponding x from 2, and substitute
back into ln(4-x) + ln(4+z) + ln(z) to find its maximum.
RonL
You are looking for the maximum of g(x,z)=ln(4-x) + ln(4+z) + ln(z), but:
g(x,z)=ln( (4-x)(4+x)x )
by the law of logarithms.
Then let:
f(x,z)=exp(g(x,z))=(4-x)(4+x)x,
and because the exponential function is increasing on the real numbers, any
maxima of g(x,z) is also a maxima of f(x,z) and vice-versa.
RonL