Stationary points (roots) of partially derivated function

[brollysan]

Homework Statement

f(x,y)= ln(x+y) -x^2 - y^2

1. Find the partially derivatives
2. Find the stationary points (roots) of the function

Homework Equations

The Attempt at a Solution

Quite simple, except i don't know what to do with the ln part, this is my attempt tho

f'x= 1/(x+y) -2x
f'y= 1/(x+y) -2y

Is this right? I am not sure what the derivative of ln (x+ c) is c= constant

2. Roots= where the function crosses the x-line right? So I simply set f'x=0 and solve it to get the root?

 

[Dick]

f'x and f'y are correct. The derivative of ln(f(x)) is f'(x)/f(x). They then want values of x and y where BOTH f'x and f'y equal zero simultaneously. These are the stationary points. Can you find them?

 

[brollysan]

If they both equal zero then they must both equal each other right? Then that means x=y?

F'x=0 = 1/(2x) -2x => 4x^2 =1 => x= 1/4

And since x = y then they must both have a root of 1/4...am i right?

 

[brollysan]

While i am at it i also need help with a limit problem.

The problem, all variables etc

f(x)= x^2 + 2x

find

lim x->a [f(x) - f(a)]/ (x-a)

The attempt at a solution

The correct answer is 2a +2

Factoring etc i got this far:

[(x-a)(x+a) -2a -2x]/(x-a)

Now i can write this as [(x-a)(x+a)]/(x-a) + -2a-2x / (x-a)

Leaving me with x+a -2a-2x/(x-a) I am stuck with this, how do make it so the last part simply becomes 2?

 

[Dick]

If they both equal zero then they must both equal each other right? Then that means x=y?

F'x=0 = 1/(2x) -2x => 4x^2 =1 => x= 1/4

And since x = y then they must both have a root of 1/4...am i right?

No. x^2=1/4, not x=1/4. And remember quadratic equations generally have two roots. What are the (x,y) pairs that satisfy this?

 

[Dick]

While i am at it i also need help with a limit problem.

The problem, all variables etc

f(x)= x^2 + 2x

find

lim x->a [f(x) - f(a)]/ (x-a)

The attempt at a solution

The correct answer is 2a +2

Factoring etc i got this far:

[(x-a)(x+a) -2a -2x]/(x-a)

Now i can write this as [(x-a)(x+a)]/(x-a) + -2a-2x / (x-a)

Leaving me with x+a -2a-2x/(x-a) I am stuck with this, how do make it so the last part simply becomes 2?

Surely you can simplify (-2a-2x)/(x-a). (x-a) is a factor of the numerator.