Download Finding Local Extrema and Saddle Points with Derivative Tests and more Assignments Computer Science in PDF only on Docsity!
"In preparation of these slides, materials have been taken
from different online sources in the shape of books,
websites, research papers and presentations etc. However,
the author does not have any intention to take any benefit of
these in her/his own name. This lecture (audio, video, slides
etc) is prepared and delivered only for educational purposes
and is not intended to infringe upon the copyrighted
material. Sources have been acknowledged where applicable.
The views expressed are presenter’s alone and do not
necessarily represent actual author(s) or the institution."
Extreme values and saddle point (14.7)
Lagrange Multipliers (14.8)
Group Members:
Alishba Jamil (BCS-001)
Kainat Ahmad (BCS-016)
Arooj Zafar (BCS-005)
Joveria Raja (BCS-068)
LOCAL MAXIMA
F(Xo,Yo)>f(x,y)
LOCAL MINIMA
F(Xo,Yo)<f(x,y)
ABSOLUTE MAXIMA
If preceding inequality holds For all domain of the function.
ABSOLUTE MINIMA
If the preceding inequality holds for all domain of the function
SECOND DERIVATIVE TEST:
Suppose that f(x,y) an its first and second partial derivatives are continuous
throughout a disk centered at (a,b) and that fx(a,b)=fy(a,b)=o then,
f has a local max at (a,b) if fxx<0 and
fxx.fyy-fxy
2
0 at (a,b)
f has a local min at (a,b) if fxx>0 and
fxx.fyy-fxy
2
0 at (a,b)
f has a saddle point at (a,b) if fxx.fyy-fxy
2
<
Test is inconclusive if fxx.fyy-fxy
2
=
Local minima
- Find fx then fxx
- Find fy then fyy
- Then find critical point
- Then we find fxy
2
- put all values in (fxx)(fyy)-fxy
2
- then check the condition of local maxima that is if fxx>0 and
fxx.fyy-fxy
2
0 at (a,b)
CRITICAL POINT:
fx(x,y)=fy(x,y)=
Either fx(x,y) or fy(x,y) does not exist
A saddle point or minimax point is a point on the surface of the
graph of a function where the slopes (derivatives) in orthogonal
directions are all zero (a critical point), but which is not a local
extremum of the function.
Hessian/discriminant
The expression (fxx)(fyy)-fxy
is called the hessian or discriminant of f .It is
sometimes easier to remember it in determinant form,
(fxx)(fyy)-fxy
2
fxx fxy
fxy fyy
Lagrange Multipliers
Sometimes we need to find the extreme values of
a function whose domain is constrained to lie
within some particular subset of the plane.it is the
strategy for finding local maxima and minima of
a function.
f = λ g
λ is the Lagrange multiplier
Steps
- (^) Find the partial derivative of f(x,y,z)
- Comparing f
x
= λg x
, f y
= λg y
, f z
= λg z
