Local Extrema
- Author:
- Andreas Lindner
- Topic:
- Calculus
The Hesse matrix ist given by
Theorem
Let be a twice differentiable function with grad f(x0,y0) = 0 and be a symmetric matrix then
i) det Hess > 0 ∧ a1,1 > 0 ⇒ Hess is positive definite, i. e. in (x0,y0) is a local minimum of f.
ii) det Hess > 0 ∧ a1,1 < 0 ⇒ Hess is negative definite, i. e. in (x0,y0) is a local maximum of f.
iii) det Hess < 0 ⇒ Hess is indefinite, i. e. in (x0,y0) is a saddle point.
Task
Move point P and try to find maxima, minima or saddle points.
Use functions such as
f(x,y) = x*y , f(x,y) = 0.5(x³ + x² - x) - 0.5y² , f(x,y) = sin(x)*sin(y) etc.
Hint: You can change the properties of the grid in the xy-plane to distance π or π /2 as needed.