Final answer:
To find the points where the tangent line is horizontal, find the derivative and set it equal to zero. Solve for x and substitute it back into the original equation to find the values of y. The solutions to y³ - 3y² = 1 are the points where the tangent line is horizontal.
Step-by-step explanation:
To find the points where the tangent line is horizontal for the equation x²+xy+y²=1, we need to find the derivative and set it equal to zero. The derivative of the equation with respect to x is 2x+y. Setting this derivative equal to zero, we get 2x+y=0. Solving for x, we find that x = -y/2.
Now, substitute this value of x back into the original equation to find the values of y when the tangent line is horizontal. We have (-y/2)² + (-y/2)*y + y² = 1. Simplifying this equation, we get y³ - 3y² = 1.
The points where the tangent line is horizontal are the solutions to this equation. You can solve it graphically or numerically using a calculator to find the approximate values of y.