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. |