Question

Let f

be a function of two variables. Which of the following statements is true?
Select one:
a. If (a,b)
is a critical point of f, then f has a local maximum at (a,b)
.
b. f
must have a saddle point.
c. If f
has a local maximum at (a,b), then (a,b)
is a critical point.
d. If (a,b)
is a critical point of f, then f has a local minimum at (a,b).

Answer 1

c)If f

has a local maximum at (a,b), then (a,b)

is a critical point. 

In other statements the sufficient condition of the extremum of the function of two variables is not fulfilled

Theorem: [The Second Derivative Test for Local Extreme Values]

Suppose that $f(x,y)$ has continuous second order partial derivatives at $(a,b)$ and also that

$f_x(a,b)=0=f_y(a,b).$ Let

$\Delta=\begin{vmatrix} f_{xx}& f_{xy} \\ f_{yx}&f_{yy} \end{vmatrix}$

Then:

• f has a local minimum at (a,b) if ∆(a,b) > 0 and

$f_{xx}(a,b)>0;$ .

• f has a local maximum at (a,b) if ∆(a,b) > 0 and $f_{xx}(a,b)<0;$

• f has a saddle point at (a,b) if ∆(a,b) < 0