Skip to content
Sahithyan's S2
Sahithyan's S2 — Methods of Mathematics

Extremums

Revise Extremums from S1.

Theorems

Local extremum

Suppose ff has a local maximum or minimum at (a,b)(a,b) and the first order partial derivatives of ff exist there. Then fx(a,b)=0f_x (a, b) = 0 and fy(a,b)=0f_y (a, b) = 0.

Extreme Value Theorem

If ff is continuous on a closed, bounded set DR2D \subseteq \mathbb{R}^2 then ff has an absolute maximum and absolute minimum value in DD.

Implicit Function Theorem

Suppose f(x,y)C1f(x, y) ∈ C^1, fy(a,b)0f_y(a, b) \neq 0 and f(a,b)=cf(a, b) = c. Then there exists a unique function y=g(x)C1y = g(x) ∈ C^1 defined on a neighbourhood of (a,b)(a, b) with:

g(x)=fx(x,g(x))fy(x,g(x))    s.t.    f(x,g(x))=cg'(x) = − \frac{f_x(x,g(x))} {f_y(x,g(x))}\;\; \text{s.t.}\;\; f(x, g(x)) = c

Second Derivatives Test

Suppose the second partial derivatives of ff are continuous on a disk with center (a,b)(a, b), and suppose that fx(a,b)=0f_x (a, b) = 0 and fy(a,b)=0f_y (a, b) = 0. Let:

D=[fxxfxyfyxfyy]=D(a,b)=fxx(a,b)fyy(a,b)fxy2(a,b)D = \begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix} = D(a, b) = f_{xx}(a, b)f_{yy} (a, b) − f_{xy}^2(a, b)
  • If D=0D=0, then no information.
  • If D>0fxx(a,b)>0D > 0 \land f_{xx} (a, b) > 0, then f(a,b)f(a, b) is a local minimum.
  • If D>0fxx(a,b)<0D > 0 \land f_{xx} (a, b) < 0, then f(a,b)f(a, b) is a local maximum.
  • If D<0D < 0 then f(a,b)f(a, b) is not a local maximum or minimum.

Saddle point

A critical point in which it is not a local maximum or local minimum.

Closed interval method

A method for finding absolute minimum and absolute maximum values of ff which is continuous on a closed, bounded set DR2D \subseteq \mathbb{R}^2.

  1. Find the values of ff at the critical points of ff in the interior of DD.
  2. Find the extreme values of ff on the boundary of DD.
  3. Maximum and minimum values from step 1 and 2 are the absolute maximum and absolute minimum.