Question

how to determine the maxima, minima or saddle on partial derivative?

Answer 1

there is a rule that implements partial derivatives:

Fxx Fyy - (Fxy)^2

this value is similar to a determinant in a quadratic equation and its output tells us about saddle points or not

Answer 2

D < 0, saddle point
D = 0, undetermined
D > 0, observe Fxx
Fxx is +, min
Fxx is -, max

Answer 3

i've already got the result of that...
the result of Fxx Fyy - F(xy)^2 is bigger than zero...
but the Fxx is zero...
so how to determine it?

Answer 4

youve made an error then

0 - F^2 is not bigger than 0 ... unless you have some complex values i spose

Answer 5

hahaha...
how careless am i...
thx for your correction sir....

Answer 6

Suppose \(f(x)\) is a function of x that is twice differentiable at a stationary point \(x_0\).

1. If \(f^{''}(x_0)>0\), then \(f\) has a local minimum at \(x_0\).

2. If \(f^{''}(x_0)<0\), then \(f\) has a local maximum at \(x_0\).

The extremum test gives slightly more general conditions under which a function with \(f^{''}(x_0)=0\) is a maximum or minimum.

If f(x,y) is a two-dimensional function that has a local extremum at a point \((x_0,y_0)\) and has continuous partial derivatives at this point, then \(f_x(x_0,y_0)=0\) and \(f_y(x_0,y_0)=0\). The second partial derivatives test classifies the point as a local maximum or local minimum.

Define the second derivative test discriminant as

\(~~~~D \equiv f_{xx}f_{yy}-f_{xy}f_{yx}\)
\(= f_{xx}f_{yy}-f_{xy}^2.\)

Then

1. If \(D>0\) and \(f_{xx}(x_0,y_0)>0\), the point is a local minimum.

2. If \(D>0\) and \(f_{xx}(x_0,y_0)<0\), the point is a local maximum.

3. If \(D<0\), the point is a saddle point.

4. If \(D=0\), higher order tests must be used.

Answer 7

youre welcome

Answer 8

one way i recall the min/max of second derivatives is to consider the function x^2

f' = 2x
f'' = 2

i know that x^2 has a minimum, and that 2 is positive ...

if f'' is positive, then caveUP is a good visual