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.