To find the stationary points on the curve given by the equation, we first need to find the derivative of this function with respect to the variable, in this case, y.
The given equation is (x + y - 2)^2 = e^y - 1
1. Express the equation in the form f(x, y) = 0 and rewrite the equation.
`(x + y - 2)^2 - e^y + 1 = 0`
2. Differentiate both sides of the equation implicitly with respect to y.
Using the chain rule for differentiation,
`(2*(x + y - 2) * (1 + dy/dx)) - e^y = 0`
Here, dy/dx represents the first derivative of y with respect to x.
3. Solving for dy/dx results in an equation that provides the slope of the function at any point (x, y).
`dy/dx = [e^y - 2*(x + y - 2)] / 2*(x + y - 2)`
4. The stationary point(s) on the curve occur at dy/dx = 0. That is when the numerator of the above equation is zero.
`0 = e^y - 2*(x + y - 2)`
5. The solutions to this equation will give the y-values of the stationary points. In order to find these y-values, we need to solve the complex equation
`e^y = 2*(x + y - 2)`
6. The x-values corresponding to these y-values can be found by substituting the y-values back into the original equation. Then we can solve the equation for x.
Let's note that it's quite difficult to solve this equation analytically and it might require numerical methods, such as the Newton-Raphson method, to find the exact coordinates of the stationary points.
Please remember that not every point where dy/dx = 0 will be a stationary point, this needs to be checked. A point is a stationary point if the function changes its slope around this point from positive to negative or vice versa. A point where dy/dx = 0 and where the function does not change the sign of its slope is a point of inflection. To distinguish between these, we calculate the second derivative, d^2y/dx^2, and substitute the point coordinates there. If d^2y/dx^2 < 0, we have a local maximum, if d^2y/dx^2 > 0, we have a local minimum. If we calculate the d^2y/dx^2 and it equals to zero, we can't conclude anything and need to use other methods to clarify the nature of the point.