To find: The points on the lemniscate where the tangent is horizontal.
The points on the lemniscate where the tangent is horizontal are .
The equation is (1)
Derivative rule: Chain rule
If and are both differentiable function, then .
Obtain the point at which the tangent line is horizontal.
Consider the equation .
Differentiate implicitly with respect to x,
Apply the chain rule (1) and simplify the terms,
Combine the terms to one side of the equation,
Thus, the derivative of the equation is .
Note that, the tangent is horizontal if .
Substitute in equation (1),
Simplify the terms and obtain the value of y,
Substitute in ,
Therefore, the tangent line is horizontal at the points .
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