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\textbf{Applications of Partial Differentiation}

\textit{\textbf{Extremes}}
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\textbf{Question}

Show that there exists a local maximum point of the function
$$f(x,y,z)=4xyz-x^4-y^4-z^4$$ 
at $(1,1,1)$.


\textbf{Answer}

\begin{eqnarray*}
D & = & f(1+h,1+k,1+m)- f(1,1,1)\\
& = & 4(1+h)(1+k)(1+m)-(1+h)^4-(1+k)^4-(1+m)^4-1\\
& = & 4(1+h+k+m+hk+hm+km+hkm)\\
& & -(1+4h+6h^2+4h^3+h^4)\\
& & -(1+4k+6k^2+4k^3+k^4)\\
& & -*1+4m+6m^2+4m^3+m^4)-1\\
& = & 4(hk +hm+km)- 6(h^2+k^2+m^2)+\cdots
\end{eqnarray*}
where $\cdots$ stands for terms of degree 3 and 4 in the variables
$h$, $k$ and $m$. Completing some squares with the quadratic terms
leads us to
$$D= -2[(h-k)^2+(k-m)^2+(h-m)^2+h^2+k^2+m^2]+\cdots$$
which is negative if $|h|$, $|k|$ and $|m|$ are small and not all
$0$. (This is due to the fact that the terms of degree 3 and 4 are
smaller in size than the quadratic terms when the variables are
small.)

Hence $f$ has a local maximum value at $(1,1,1)$.

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