What is the area of a circular wading pool with a radius of 50 cm?
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Use the method of Lagrange multipliers. We have Lagrangian
with partial derivatives (set equal to 0) of
As
, and 
, we can obtain
From this, we find a single critical point:
At this point, we have a value of
To determine what kind of extremum occurs at this point, we check the Hessian of
:
We observe that
at any point 
, and that the eigenvalues of this matrix are all positive (2 with multiplicity 3), so 
is positive definite. By the second partial derivative test, this means attains a minimum at this critical point. Meanwhile, 
has no maximum value.
with partial derivatives (set equal to 0) of
As
From this, we find a single critical point:
At this point, we have a value of
To determine what kind of extremum occurs at this point, we check the Hessian of
We observe that