Finding points with vertical tangents  Summary and Q&A
TL;DR
Use implicit differentiation to find the derivative of y with respect to x, then solve for dy/dx. Determine the y values that make the denominator equal zero to find the corresponding x values. The two points on the curve with a vertical tangent line are (4, 1) and (2, 1).
Questions & Answers
Q: How can we find the coordinates of the points on the closed curve with a vertical tangent line?
To find the points, we can use implicit differentiation to obtain the derivative of y with respect to x. By setting the denominator of the derivative equal to zero, we get the y values. Substituting these y values back into the original equation will solve for the corresponding x values.
Q: What is the process of implicit differentiation?
Implicit differentiation involves differentiating both sides of an equation with respect to x while treating y as an implicit function of x. Apply the chain rule as necessary, then solve for the derivative of y with respect to x.
Q: How do we determine the y values that make the denominator of the derivative equal to zero?
By setting the denominator of the derivative expression equal to zero, we can solve for y. These y values correspond to the points on the curve with a vertical tangent line.
Q: What are the coordinates of the points on the curve with a vertical tangent line?
The two points are (4, 1) and (2, 1). These are the x and y coordinates that satisfy the equation and have a vertical tangent line.
Summary & Key Takeaways

The task is to find the coordinates of the two points on a closed curve where the tangent line is vertical.

Implicit differentiation is used to find the derivative of y with respect to x.

By setting the denominator of the derivative equal to zero, the y values that correspond to the points with a vertical tangent line are found.

Substitute these y values back into the original equation to solve for the corresponding x values.