What are the values and types of the critical points, if any, of #f(x, y) = xy^2+xy-3x^2y-7x#?
Saddle points at
We have a 3D surface defined by:
# f(x, y) = xy^2 + xy - 3x^2y - 7x #
Step 1 - Find the Partial Derivatives
We compute the partial derivative of a function of two or more variables by differentiating wrt one variable, whilst the other variables are treated as constant. Thus:
The First Derivatives are:
# f_x \ = (partial f) / (partial x) \ \ = y^2+y-6xy -7 #
# f_y \ = (partial f) / (partial y) \ \ = 2xy+x-3x^2 #
The Second Derivatives are:
# f_(x x) =(partial^2 f) / (partial x^2) = -6y #
# f_(yy) = (partial^2 f) / (partial y^2) = 2x #
The Second Partial Cross-Derivatives are:
# f_(xy) =(partial^2 f) / (partial x partial y) =2y+1-6x #
# f_(yx) = (partial^2 f) / (partial y partial x) =2y+1-6x #
Note that the second partial cross derivatives are identical due to the continuity of
Step 2 - Identify Critical Points
A critical point occurs at a simultaneous solution of
# f_x = f_y = 0 iff (partial f) / (partial x) = (partial f) / (partial y) = 0#
i.e, when:
# y^2+y-6xy -7 = 0# ..... [A]
# 2xy+x-3x^2 = 0# ..... [B]From [B], we have:
# x(2y+1-3x) = 0 => x=0,1/3(2y+1) # From [A], we have:
# x=0 => y^2+y -7 = 0 #
# :. y = -1/2+-sqrt(29)/2 #
# x=1/3(2y+1) => y^2+y-6(1/3(2y+1))y -7 = 0 #
# :. y^2+y-4y^2-2y -7 = 0 #
# :. 3y^2+y +7 = 0 # having no real solutionsSo we can conclude that there are two critical points:
# (0,-1/2-sqrt(29)/2) # and#(0,-1/2+sqrt(29)/2) # Step 3 - Classify the critical points
In order to classify the critical points we perform a test similar to that of one variable calculus using the second partial derivatives and the Hessian Matrix.
# Delta = H f(x,y) = | ( f_(x x) \ \ f_(xy) ) , (f_(yx) \ \ f_(yy)) | = | ((partial^2 f) / (partial x^2),(partial^2 f) / (partial x partial y)), ((partial^2 f) / (partial y partial x), (partial^2 f) / (partial y^2)) | = f_(x x)f_(yy)-(f_(xy))^2 # Then depending upon the value of
#Delta# :
# {: (Delta>0, "There is maximum if " f_(x x)<0),(, "and a minimum if " f_(x x)>0), (Delta<0, "there is a saddle point"), (Delta=0, "Further analysis is necessary") :} # Using custom excel macros the function values along with the partial derivative values are computed as follows:
And we can confirm these results graphically
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To find the critical points of ( f(x, y) = xy^2 + xy - 3x^2y - 7x ), we need to find where the partial derivatives with respect to ( x ) and ( y ) are both equal to zero.
The partial derivatives are: [ \frac{\partial f}{\partial x} = y^2 - 6xy - 7 ] [ \frac{\partial f}{\partial y} = 2xy + x - 3x^2 ]
Setting both partial derivatives equal to zero, we have the system of equations: [ y^2 - 6xy - 7 = 0 ] [ 2xy + x - 3x^2 = 0 ]
Solving this system of equations will give us the critical points.
From the first equation: [ y^2 = 6xy + 7 ] [ y = \pm \sqrt{6xy + 7} ]
Substitute ( y ) into the second equation: [ 2x(\pm \sqrt{6xy + 7}) + x - 3x^2 = 0 ] [ 2x^2(\pm \sqrt{6xy + 7}) + x - 3x^2 = 0 ]
This equation does not easily yield a simple solution. However, you can solve it numerically or by using computer software to find the critical points. Once the critical points are found, you can classify them by determining the type of critical point using the second partial derivative test or other methods.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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