A pendulum's first swing has an arc of 24 inches. As the pendulum swings back and forth, the arc length for each successive swin
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Take partial derivatives and set them equal to 0:
We find one critical point within the boundary of the disk at
. The Hessian matrix for this function is
which is positive definite, and incidentally independent of
and 
, so 
attains a minimum 
.
Meanwhile, we can parameterize the boundary by
with 
, which gives
with critical points at
At these points, we get
so we attain a maximum only when
, which translates to 
.
We find one critical point within the boundary of the disk at
which is positive definite, and incidentally independent of
Meanwhile, we can parameterize the boundary by
with critical points at
At these points, we get
so we attain a maximum only when