Consider the function f(x,y)=yxy24x+15y. Find and classify all critical points o
ID: 2888606 • Letter: C
Question
Consider the function
f(x,y)=yxy24x+15y.
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)
Explanation / Answer
f(x,y)= y?x?y²?4x+15y.
Find partial derivative as
fx= y/(2?x) -4
fy = ?x -2y +15
Find second partial derivative as
fxx = -y/(4*x^(3/2))
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To find critical point solve
fx= 0 and fy =0
y/(2?x) -4 =0, gives y= 8?x
?x -2y +15 = 0, plug in y= 8?x
?x -16?x +15 = 0
-15?x +15 = 0
15?x =15
?x = 1
x =1
hence, y=8?1 = 8
Thus, only critical point is (1,8)
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The critical point with smallest coordinate is
(1, 8) and classification is Local maximum
Leave the remaining all boxes Blank.