Consider the function f(x,y)=y(x)^(1/2)-y^2-5x+19y Find and classify all critica
ID: 2847343 • Letter: C
Question
Consider the function f(x,y)=y(x)^(1/2)-y^2-5x+19y
Find and classify all critical points of the function. If there are more blanks than critical points, leave the remaining entries blank.
fx=
fy=
fxx=
fxy=
fyy=
The critical point with the smallest x-coordinate is
( , ) Classification: (local minimum, local maximum, saddle point, cannot be determined)
The critical point with the next smallest x-coordinate is
( , ) Classification: (local minimum, local maximum, saddle point, cannot be determined)
The critical point with the next smallest x-coordinate is
( , ) Classification: (local minimum, local maximum, saddle point, cannot be determined)
Explanation / Answer
(a)
fx = y/2(x)^0.5 -5
(b)
fy = x^0.5 -2y +19
(c)
fxx = -y/(4*x^1.5)
(d)
fxy = 1/2(x)^0.5
fyy = -2
critical points are:
fx =0, fy = 0
=>
y = 10x^0.5, x^0.5 -2y +19 =0 => y/10 -2y +19 = 0 => y = 10
=>x = 1
(1,10 is the critical point)
D = (-10/4)(-2)-(0.5) = 4.5>0 , fxx <0 =>
(1,10) is local maximum