y2-10-Sy + 1 (1 point) Suppose f(x,y) -x2 + (A) How many critical points does f
ID: 2888123 • Letter: Y
Question
y2-10-Sy + 1 (1 point) Suppose f(x,y) -x2 + (A) How many critical points does f have in R2? (B) If there is a local minimum, what is the value of the discriminant D at that point? If there is none, type N (C) If there is a local maximum, what is the value of the discriminant D at that point? If there is none, type N (D) If there is a saddle point, what is the value of the discriminant D at that point? If there is none, type N. at is the maximum value on (F) What is the minimum value of f on R2? If there is none, type NExplanation / Answer
Given f(x,y) = x^2 + y^2 - 10 x - 8 y + 1
fx = /x ( x^2 + y^2 -10x - 8 y + 1 ) and fy = /y ( x^2 + y^2 - 10x - 8y + 1
= 2x - 10 = 2y - 8
fxx = /x ( 2x - 10 ) and fyy = /y ( 2y - 8 )
= 2 = 2
fxy = /y ( 2x - 10 ) and fyx = /x ( 2y - 8 )
= 0 = 0
for critical Points
fx = 0 and fy = 0
2 x - 10 = 0 2y - 8 = 0
2x = 10 2 y = 8
x = 5 y = 4
so ( 5 , 4 ) is only critical Point
Second derivative Test :
D(a,b) = fxx(a,b) * fyy( a, b) - [ fxy(a,b) * fyx(a,b) ]
If D >0 and fxx(a, b) >0, then f(a, b) is a local minimum
If D >0 and fxx(a, b) < 0, then f(a, b) is a local maximum
If D < 0 , then (a, b) is a saddle point
D( 5 , 4 ) = fxx( 5 , 4 ) * fyy( 5 , 4 ) - [ fxy( 5 , 4 ) * fyx( 5 , 4 ) ]
= 2 * 2 - [ 0 * 0 ]
= 4 > 0
fxx( 5 , 4 ) = 2 > 0
so
f(5,4) = 5^2 + 4^2 - 10 (5) - 8 (4) + 1
= 25 + 16 - 50 - 32 + 1
f(5,4) = - 40( minimum value ) is a local minimum
( A )
Critical Point is ( 5,4 )
( B )
Local minimum , D = 4
( C )
N
( D )
N
( E )
N
( F )
f (5,4) = - 40 ( minimum value )
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