How do you find an equation for the horizontal tangent plane to the surface #z=4(x-1)^2+3(y+1)^2#
1 Answer
The answer is:
Remember that an horizontal plane is tangent to a curve in the space in its points of maximum, minimum or saddle.
We can answer in two ways.
The first: this function is the equation of an elliptic paraboloid with concavity upwards. Since
so the vertex is
The second is really long, but it's the only way if you don't recognize the curve.
We have to calculate the two partial derivative:
The system of the two previous equations (put
so:
and we can calculate the applicate
Now we have to find what type of stationary point
To do that we have to calculate the four partial derivative of the second order:
And the Hessian matrix:
Since its determinant in