Mari has a part time job. She earns $7 an hour. She makes at most $143.50 a week. What is the greatest number of hours that she
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Compute the gradient of .
Set this equal to the zero vector and solve for the critical points.
The last case has no real solution, so we can ignore it.
Now,
so we have two critical points (0, 0) and (2, 2).
Compute the Hessian matrix (i.e. Jacobian of the gradient).
Check the sign of the determinant of the Hessian at each of the critical points.
which indicates a saddle point at (0, 0);
We also have , which together indicate a local minimum at (2, 2).