Ben LoBrutto

Footnote 18 Project

12:40 Class

11.7 DISCOVERY PROJECT: Quadratic Approximations and Critical Points

In Section 11.4 the linearization of a function f of two variables at point (a, b)  is defined as:

The graph of L is the tangent plane to the surface z = f(x, y) at (a, b, f(a, b) and the corresponding linear approximation is f(x, y) = L(x, y).  The Linearization L is also called the first degree Taylor polynomial of f at (a, b).

Step 1.    If f has continuous second-order partial derivatives at (a, b) , then the second-degree Taylor polynomial of f at (a, b) is

And the approximation

is called the quadratic approximation to f  at (a, b).


Verify that Q has the same first- and second- order partial derivatives as f at (a, b ).

Since f (a, b) is a constant:

=

Now,

Qx = f_x(a, b) + f_xx(a, b) x 0 + f_yy(a, b) x 0

EQUAL

EQUAL

Step 2. In this problem the behavior of the polynomial  f(x, y) = ax² + bxy + cy² is analyzed by identifying the graph as a paraboloid.

EQUAL

b)      Let  D = 4ac - b² .  Show that if D > 0 and a > 0, then f  has a local minimum at (0,0).

Remember:

Local min when:

So, if  a > 0, and 4ac - b² > 0, then f_xx > 0, and D > 0 , so  f  has a local minimum at (0,0)

c) Show that if D > 0 and a < 0, then  f   has a local maximum at (0,0)

Local max when D > 0, 0

If  D > 0, then 4ac - b²   is positive. So,

(+

So, f has a local max at (0,0).

Saddle point when D < 0

If 4ac - b² < 0, then two scenarios can exist:

a is positive, in which case:

Or, a is negative, in which case:

So, if  D < 0, then D is negative, and is a saddle point.  



Step 3.  a) Suppose is any function with continuous second-order partial derivatives such that  f (0, 0) = 0, and (0,0) is a critical point of  f .

Write an expression for the second-degree Taylor polynomial  Q , of  f  at (0,0).

b) What can you conclude about  Q from Problem 2?

From Problem 2 we can conclude that:

is positive, and

is negative, and

If   is negative, then  (0, 0)  is a saddle point.

c) In view of the quadratic approximation , what does part (b) suggest about  f ?

Part (b) suggests that f has either a local min, local max, or saddle point at (0,0)