Implicit Differentiation

[Vempy]

For the lemniscate 2(x^2+y^2)^2 = 25(x^2-y^2) find the points where the tangent is horizontal.

The derivative is:

dy/dx= x(25-4(x^2-y^2))
y(25+4(x^2+y^2))

How do I find the points where the tangent is horizontal? I don't even know where to start. :shock:

 

[skeeter]

horizontal tangent means \(\displaystyle \L \frac{dy}{dx} = 0\)

set the numerator of the derivative = 0 ...

\(\displaystyle \L x[25-4(x^2-y^2)] = 0\)

\(\displaystyle \L x = 0\) or \(\displaystyle \L x^2 = y^2 + \frac{25}{4}\)

using the original equation of the curve ...
if \(\displaystyle \L x = 0\), then \(\displaystyle \L 2y^4 = -25y^2\)

\(\displaystyle \L 2y^4 + 25y^2 = 0\)

\(\displaystyle \L y^2(2y^2 + 25) = 0\)

\(\displaystyle \L y = 0\) ... the point (0,0) is on the curve and will have a horizontal tangent.

now for the other possible ... substitute \(\displaystyle \L y^2 + \frac{25}{4}\) for \(\displaystyle \L x^2\) in the original equation ... solve for y to find any other possible points on the curve that have a horizontal tangent.

I'll leave that part to you.