Let f(x, y) = x2 + 3xy + 5y2 + 4x3 - y3. Use the second derivative test to decid
ID: 2868176 • Letter: L
Question
Let f(x, y) = x2 + 3xy + 5y2 + 4x3 - y3. Use the second derivative test to decide if the point (0.0) is a local minimum or local maximum or a saddle point. If the test is inconclusive, explain why. Let f(x, y) = X2 +3xy - 5y2 + x3 -3y3. Use the second derivative test to decide if the point (0,0) is a local minimum or local maximum or a saddle point. If the test is inconclusive, explain why. Let f(x,y) = -2x2 + 7xy - 10y2 + x3 - 3y3. Use the second derivative test to decide if the point (0,0) is a local minimum or local maximum or a saddle point. If the test is inconclusive, explain why. Use Lagrange Multipliers to calculate the minimum value of the function the surface 2x + y + 2z = 9.Explanation / Answer
a)
f(x,y) = x^2 + 3xy + 5y^2 + 4x^3 - y^3
fx = 2x + 3y + 12x^2
fxx = 2 + 24x
At (0,0), it is : 2 + 24(0) --> fxx = 2
fy = 3x + 10y - 3y^2
fyy = 10 - 6y
At (0,0), it is : 10 - 6(0) --> fyy = 10
fxy = 3
D = fxx*fyy - (fxy)^2
D = 2(10) - 3^2
D = 20 - 9 = 11
Since D > 0 and fxx > 0, (0 , 0) is a LOCAL MINIMUM----> ANSWER
---------------------------------------------------------------------------------------------------
b)
f = x^2 + 3xy - 5y^2 + x^3 - 3y^3
fx = 2x + 3y + 3x^2
fxx = 2 + 6x
At (0,0), fxx = 2 + 6(0) = 2
fy = 3x - 10y - 9y^2
fyy = -10 - 18y
At (0,0), fyy = -10 - 18(0) = -10
fxy = 3
D = fxx*fyy - fxy^2
D = 2(-10) - 3^2
D = -20 - 9
D = -29
Since D < 0, (0,0) is a SADDLE POINT ----> ANSWER
--------------------------------------------------------------------------------------------------
c)
f = -2x^2 + 7xy - 10y^2 + x^3 - 3y^3
fx = -4x + 7y + 3x^2
fxx = -4 + 6x
At (0,0), fxx = -4 + 6(0) = -4
fy = 7x - 20y - 9y^2
fyy = -20 - 18y
At (0,0), fyy = -20 - 18(0) = -20
fxy = 7
D = fxxfyy - fxy^2
D = (-4)(-20) - (7)^2
D = 80 - 49
D = 31
Since, D > 0 and fxx < 0, (0 , 0) is a LOCAL MAXIMUM
------------------------------------------------------------------------------------------------------------
d)
f(x,y,z) = x^2 + y^2 + z^2
constraint ---> g(x,y,z) : 2x + y + 2z = 9
fx = 2x , fy = 2y , fz = 2z
gx = 2 , gy = 1 , gz = 2
fx = m*gx
2x = 2m
x = m
fy = m*gy
2y = m*1
y = m/2
fz = m*gz
2z = m*2
z = m
So, plug these into the constraint :
2x + y + 2z = 9
becomes
2m + m/2 + 2m = 9
9m/2 = 9
m = 2
So, x = m = 2 , y =m/2 = 1 , z = m = 2
x^2 + y^2 + z^2 ---> 2^2 + 1^2 + 2^2 ---> 4 + 1 + 4
9 ----> ANSWER
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.