# What are extrema and saddle points of #f(x,y)=(x+y+1)^2/(x^2+y^2+1)#?

We have:

# f(x,y) = (x+y+1)^2/(x^2+y^2+1) #

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 = { (x^2+y^2+1)(2(x+y+1)) - ((x+y+1)^2)(2x) } / (x^2+y^2+1)^2 #

# \ \ \ = { 2(x^2+y^2+1)(x+y+1) - 2x(x+y+1)^2 } / (x^2+y^2+1)^2 #

# \ \ \ = { 2(x+y+1)(x^2+y^2+1- x^2-xy-x) } / (x^2+y^2+1)^2 #

# \ \ \ = { 2(x+y+1)(y^2-xy-x+1) } / (x^2+y^2+1)^2 #

# f_y = { (x^2+y^2+1)(2(x+y+1)) - ((x+y+1)^2)(2y) } / (x^2+y^2+1)^2 #

# \ \ \ = { 2(x^2+y^2+1)(x+y+1) - 2y(x+y+1)^2 } / (x^2+y^2+1)^2 #

# \ \ \ = { 2(x+y+1)(x^2+y^2+1- y^2-xy-y) } / (x^2+y^2+1)^2 #

# \ \ \ = { 2(x+y+1)(x^2-xy-y+1) } / (x^2+y^2+1)^2 #

The Second Derivatives (quoted) are:

# f_(x x) = ( -4(-x^3-x^3y-3x^2y+3xy^2+3x+3xy^3+3xy+y^3+y ) ) / (x^2+y^2+1)^3 #

# f_(yy) = (-4(3x^3y+x^3+3x^2y+3xy+x-xy^3-3xy^2+3y-y^3)) / (x^2+y^2+1)^3 #

The Second Partial Cross-Derivatives are:

# f_(xy) = {2(-x^4-2x^3+6x^2y^2+6x^2y+6xy^2-2x+1-y^4-2y^3-2y)} / (x^2+y^2+1)^3 #

# f_(yx) = {2(-x^4-2x^3+6x^2y^2+6x^2y+6xy^2-2x+1-y^4-2y^3-2y)} / (x^2+y^2+1)^3 #

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:

Solving [A] and [B] simultaneously, we gain a single solution:

# x = y = 1 #

So we can conclude that there is one critical points:

# (1,1) #

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>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:

By signing up, you agree to our Terms of Service and Privacy Policy

To find the extrema and saddle points of the function ( f(x,y) = \frac{{(x + y + 1)^2}}{{x^2 + y^2 + 1}} ), we need to follow these steps:

- Find the partial derivatives of ( f(x, y) ) with respect to ( x ) and ( y ) to find critical points.
- Determine the second partial derivatives to classify critical points as local minima, local maxima, or saddle points.

First, let's find the partial derivatives:

[ \frac{{\partial f}}{{\partial x}} = \frac{{2(x + y + 1)(1)(x^2 + y^2 + 1) - (x + y + 1)^2(2x)}}{{(x^2 + y^2 + 1)^2}} ] [ \frac{{\partial f}}{{\partial y}} = \frac{{2(x + y + 1)(1)(x^2 + y^2 + 1) - (x + y + 1)^2(2y)}}{{(x^2 + y^2 + 1)^2}} ]

Next, find critical points by setting both partial derivatives equal to zero and solving for ( x ) and ( y ).

Then, calculate the second partial derivatives:

[ \frac{{\partial^2 f}}{{\partial x^2}} = \frac{{2(x^2 + y^2 + 1)^2 - 2(x + y + 1)^2(2x)^2 - 2(x + y + 1)(2)(x^2 + y^2 + 1)(2x)}}{{(x^2 + y^2 + 1)^3}} ] [ \frac{{\partial^2 f}}{{\partial y^2}} = \frac{{2(x^2 + y^2 + 1)^2 - 2(x + y + 1)^2(2y)^2 - 2(x + y + 1)(2)(x^2 + y^2 + 1)(2y)}}{{(x^2 + y^2 + 1)^3}} ] [ \frac{{\partial^2 f}}{{\partial x \partial y}} = \frac{{(x^2 + y^2 + 1)^2 - 2(x + y + 1)(2x)(2y) - (x + y + 1)^2(2)}}{{(x^2 + y^2 + 1)^3}} ]

Finally, substitute the critical points into the second partial derivatives to classify them as local minima, local maxima, or saddle points.

By signing up, you agree to our Terms of Service and Privacy Policy

- Is #f(x)=sinx/x# increasing or decreasing at #x=pi/3#?
- What are the extrema of #g(x) = cos^2x+sin^2x?# on the interval #[-pi,pi#?
- What are the local extrema, if any, of #f(x) =(lnx-1)^2 / x#?
- Is #f(x)=(x^3-5x^2-x+2)/(2x-1)# increasing or decreasing at #x=0#?
- Given the function #f(x)=sqrt(2-x)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-7,2] and find the c?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7