Find all values (k) and (m) on the curve

Question

Find all values (k) and (m) on the curve x^3-6xy-ky^3=m where the slope of the tangent line is -1 at point (0,1)

Answer 1

We need to get two equations.  One equation will be the value at the curve.  The other will contain derivative.

 

For the first equation, find the value on the curve.

 

-k = m

 

 

For the second equation, get the derivative using implicit differentiation.  Remember that m and k are constants.

 

3x² - (6y + 6xy') - 3ky²y' = 0

 

3x² - 6y - 6xy' - 3ky²y' = 0

 

 

Group all the y' terms to one side.

 

-6xy' - 3ky²y' = 6y - 3x²

 

y' (-6x - 3ky²) = 6y - 3x²

 

y' = (6y - 3x²) / (-6x - 3ky²)

 

 

Plug in the point x=0 and y=1.

 

y' = 6 / (-3k)

 

y' = -2 / k

 

 

The derivative is -1.  Therefore,

 

-1 = -2 / k

 

 

k = 2

 

Then,

 

m = -2