Consideration of this function reveals that for large positive
values of z the cubic term dominates and the function value
is a large negative number. For large negative values of z,
the cubic term again dominates and the function has a large
positive value. There is a local minimum and a local maximum
nearer to the origin.
We select the Optimize
option from the menu
and ask the program to find the minimum of H starting at 0
for the variable z. The program discovers that at z = 0, the
gradient is 10 and the normalized gradient is 1. Moving in
the direction of the greatest decrease, the line search finds
the minimum at z = -9.37783. For a single dimension, the Hessian
is simply the second derivative of the function at the stationary
point. Since it is positive, the analysis concludes that this
is indeed a local minimum