Find the points on the given curve where the tangent line is horizontal or vertical

To find the points on the curve where the tangent line is horizontal or vertical, we need to analyze the derivative of the function representing the curve.

A tangent line is horizontal where the derivative is equal to zero, and it is vertical where the derivative is undefined.

For a function expressed as y = f(x), we start by finding the derivative dy/dx. Then, we set the derivative equal to zero to find where the tangent line is horizontal:

dy/dx = 0

Next, we determine where the derivative does not exist or is undefined to identify where the tangent line is vertical:

dy/dx is undefined

After calculating the derivative, we can solve these equations to find the specific points on the curve. Each point can then be represented in coordinate format (x, y).

Make sure to substitute your findings back into the original curve equation to verify the corresponding y-values for each x-value obtained.

This method will give you all points where the tangent lines are either horizontal or vertical on the curve.