A function at (a,b) (x, y) in → Similar
if some for
f(x, y) = disk with local min.
a local fla, b)
max for all (a, b)
a local max (a, b), then
or min at If (a, b) =o.
><--- 4 = 6 We also discuss several other topics like Who is daniel shays?
(a, b) is of (a, b) =
Def. A point
- f if Note: - max /min O. C. P If you want to learn more check out What is the partial vapor pressure of each component of an ideal mixture of liquids is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture?
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☆ C.P. max/ min
of equal to o find C.P..
2nd Derivative test ( Single variable)
4= f(x), xsa is a CP. if flla) = 0 f" (a) >0 & local min. at x=a f" (a) co
local max . at x a
ex Find all local max / mins Saddle points. o f(x,y) = x + 3y’ -lexy +10x -ley +B + = 2x - (au + 10, au -ex - We also discuss several other topics like What is alkylation in chemistry?
fx = 0 = 2x -64 +10 0... 0 = -4* +4 fy = 0.64-6-6
-4 = -4* 0 + 0 :-4x = 4 5 We also discuss several other topics like What procedure would you use to separate organelles?
> x = ( 2 cp (2)
7 y = 2 JE 2 = le D(x, y) = 12-36= -24 co
> Saddle point. If you want to learn more check out What are the contributions of the chinese in san francisco?
Second Derivative Test Suppose (a,b) is a c.p. of z=f(x,y)
(ofla, b) = o)
D = det for fayl = (fas Hyy ) - (fxy)
( Find min/max
ex 2: f(x,y) = x + y + 3xy +5
of = < 3x2 – 3y , 3y - 3x >
(a.) Dla, b) >o (6-) Dla, b) o (c.) Dla, b) co (d.) Dla, b) = 0
htxx (a, b) o min (fxx (a, b) co max 7 Saddle point in 7 Inconclusive / No Information
X7-X = 0 x(x3 - 1) = 0
> x=0, x=1 => y = 0, y = 1
D= (fxx (fyy) – (fxy).
fxx = 6x
tyy = ley
D(x,y) D(0,0) = -9 co > Saddle DCL, 1) = 27 >O y check fxx (1,1)
Le(1) = 6 0 7 local min.
Steps to find absolute extrema 0.) Check conditions of EVT
closed, bounded set 1.) Find values of function at c.P's in D 2.) Find extrema of f on the boundary of D 3.) Compare the values in I and 2.
Lo largest abs. max. y smallest. abs. min,
abs. extrema of f(x,y) = x - xy +y?
- Absolute Single variable Let y =
max / min extrema
thm: f(x): If f is continuous on a
closed interval La,b], then f has an abs. man and an abs. min (Extreme value theorem)
a closed set in IR is one all its boundary points.
A bounded set in ik is one contained within some disk
o) The f is continuous 0722 on D, which is closed & I boun ded. => abs max/min (EVT 1.) C.Di
fx= 1-4 = 0 7 4 = 1 f(2,1)
y = -x + 2y =
o x =2
IC.P: (x, y) = (2,1) 2.) On 4 . f(x,y) = f(x,0)= b
- abs Since it is inc., max on a Lib.
. emin on [0, 3) = 0
If Set may
is D &
continuous on in R, then f abs. min
a closed, bounded attains an abs.
On Li f(3,y) = 3-3y + y2 =glyd a
gly)= -3 +34 f(3, 3/2) = 3-912.701912).
f(t) = filt) =
cost- sin’t. 2cos(+)(-Sin (+)) - 2 sin(t) coset) -4cosct) sin(t) = 0
> t = 1/2, 318/2, 0, it Plugin t's into f(t).
r(t) = F + tū
,o + t <3,2)
X(t) = 3+
| 4 ()= 2+ flï(t)) = 34- (3+)(2+) + (2t = 34 - 6+3 +4+?
Osts 12 = 34 -2t2 =h(t) hlt) = 3-40 =0 => t = 3/4 7 X = 174 A
y = 3/2
f1914, 312)= 4 (9)() +
f(0, 0)= 0 f(3, 2) = 1 f(3,0) = 3
exp 2: z= f(x, y) = x2 - y2
Find abs. extrema on x2 + y2 sl 0.) polynomial cont. closed, bounded set
max/min 1) vf = (2x -24 ) =0
CD: f(go)= 2.) Extrema on
0 *? +4
x= cos(t) y=sin(+)