4. Find the maximum and minimum values of f(x, y) = x + 2y on the disk x2 + y2 1

Question

4. Find the maximum and minimum values of f(x, y) = x + 2y on the disk x2 + y2 1. Find the maximum value of f(x, y) = xy-x3y2 over the square 0 s x s1,0sysi. :) 6. Find the maximum and minimum values of f(x, y) = xy(1-x-y) over the triangle with vertices (0, 0), (1,0), and (0, 1). 7. Find the maximum and minimum values of f(x, y) = sin x cos y on the closed triangular region bounded by the coordinate axes and the line x + y = 2 8. Find the maximum value of f(x, y) = sin x sin y sin(x + y) over the triangle bounded by the coordinate axes and the line

Explanation / Answer

7)

given f(x,y) = sinx cosy bounded by x=0, y=0 , x+y=2

fx(x,y) = cosx cosy,  fy(x,y) = - sinx siny

for critical points  :

fx(x,y) = cosx cosy=0 and fy(x,y) = - sinx siny=0

=>x=/2,3/2, y=/2,3/2, and x=0,,2,y=0,,2

critical points in the domain are (/2,0),(/2,),(3/2,0), (0,/2),(0,3/2),(,/2)

on the line x axis:

y=0

f(x,0) = sinx

maximum =1, minimum =-1

on the line y axis:

x=0

f(0,y) = sin0 cosy =0

only value =0

on the line x+y=2:

y=2-x

=>f(x,2-x) = sinx cos(2-x)

=>f(x,2-x) = sinx cosx

=>f(x,2-x) = (1/2)sin2x

minimum =-1/2 ,maximum =1//2

f(x,y) = sinx cosy

f (/2,0)=1,

f(/2,)=-1,

f(3/2,0)=-1,

f(0,/2)=0,

f(0,3/2)=0,

f(,/2)=0

so the absolute maximum is 1 ,absolute minimum is -1

please rate if helpful. please comment if you have any doubt

Related Questions

Navigate

• Browse (All)
• Browse 4
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.