You add a 4-foot-by-4-foot section of land to a 6-foot-by-8-fooy garden. Find the area of the new garden.
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Given 
subject to the constraint 
Let
.
The gradient vectors of
and 
are:
and 
By Lagrange's theorem, there is a number
, such that
It can be seen that has local extreme values at the given region.
Let
The gradient vectors of
By Lagrange's theorem, there is a number
It can be seen that