critical pts of f(x,y,z) = ln((x^7)*(y^11)*(z^2))-2xy-11y-7z

[dopey9]

Basically i'v been given the following function

Let f(x, y, z) = ln ( (x^7)*(y^11)*(z^2)) − 2xy − 11y − 7z, where x > 0, y > 0, z > 0.

and i want to find the values of x, y and z at any critical point(s) of this function, and hence determine any local maximum or minimum values of f(x, y, z).

i was wondering whats the best way to go about this problem..can any one possibly start me off ..if not guide into how to solve the problem thankz

 

[soroban]

Re: critical pts of f(x,y,z) = ln((x^7)*(y^11)*(z^2))-2xy-11

Hello, dopey9!

Let \(\displaystyle \:f(x, y, z) \:=\:\ln(x^7y^{11}z^2)\,-\,2xy\,-\,11y\,-\,7z,\,\) where \(\displaystyle x,y,z\,>\,0\)

Find the values of \(\displaystyle x\), \(\displaystyle y\) and \(\displaystyle z\) at any critical point(s) of this function,
and hence determine any local maximum or minimum values of \(\displaystyle f(x, y, z).\)

I would expand that log portion:

\(\displaystyle \L f(x,y,z) \;=\;7\cdot\ln x\,+\,11\cdot\ln y\,+\,2\cdot\ln z \,-\,2xy\,-\,11y\,-\,7z\)


Then solve the system of equations:

. . \(\displaystyle \L\begin{array}{ccccc}f_x & \:=\: & \frac{7}{x}\,-\,2y & \:=\: & 0 \\
f_y & = & \frac{11}{y}\,-\,2x\,-\,11 & = & 0 \\
f_z & = & \frac{2}{z}\,-\,7 & = & 0 \end{array}\)

 

[dopey9]

Thankz

Thanks