If-x-x-y-y-z-z-c-show-that-at-x-y-z-2-z-x-y-x-log-ex-1-

Question Number 74663 by TawaTawa last updated on 28/Nov/19

If x^{x} y^{y} z^{z} = c show that at x = y = z

\frac{\partial^{2} z}{\partial x \partial y} = - {(x \log ex)}^{- 1}

Answered by mind is power last updated on 28/Nov/19

\Rightarrow zlog (z) = \log (c) - xlog (x) - ylog (y)

let f (z) = zlog (z) = \log (c) - xlog (x) - ylog (y)

\Rightarrow \log (z) e^{\log (z)} = f (z)

\Rightarrow \log (z) = W (f (z)) \Rightarrow z = e^{W (f)}, W lambertfunction

W^{'} (t) = \frac{W (t)}{t (1 + W (t))}

Z = e^{W (f (x, y))}

\Rightarrow W (f) = \log (z)

\frac{\partial z}{\partial y} = \frac{\partial f}{\partial y} W^{'} (f (x, y)) e^{W (f)}

= (- \log (y) - 1) \frac{{We}^{W (f)}}{f (1 + W (f))}

= \frac{- (\log (y) + 1) .}{(1 + W (f))}

\frac{\partial^{2} Z}{\partial x \partial y} = \frac{\partial z}{\partial x} . \frac{- (\log (y) + 1)}{1 + W (f)} = (\log (y) + 1) . \frac{\frac{\partial f}{\partial x} . W^{'} (f)}{{(1 + W (f))}^{2}}

= (\log (y) + 1) . \frac{(- 1 - \log (x)) . W (f)}{f (x, y) . {1 + W (f)}^{3}}

= \frac{- (1 + \log (y)) (1 + \log (x)) . W (f (x, y))}{zlogz {(1 + \log (z))}^{3}} .

= \frac{Log (z)}{zlogz (1 + \log (x)}

= - {(z + zlog (e) (z))}^{- 1} = - {(x + xln (x))}^{- 1}

x (1 + \ln (x)) = x (\ln (xe) \Rightarrow

= - {(xln (xe))}^{- 1}

Commented by TawaTawa last updated on 28/Nov/19

Wow, God bless you sir .

Commented by mind is power last updated on 28/Nov/19

y^{'} re welcom