In welchen Punkten besitz die Kurve \(x^{3}-3 x^{2}+4 y^{2}=4\) waagerechte Tangenten?

Implicit differentiation can seem complex, but it’s a powerful tool in calculus. It allows us to find the derivative of functions not explicitly solved for one variable in terms of another. For example, given the curve equation \(x^{3} - 3x^{2} + 4y^{2} = 4\), we differentiate each term with respect to x. Be mindful of the product rule when differentiating y terms because y is a function of x. This process gets us to: \[3x^{2} - 6x + 8y \frac{dy}{dx} = 0\] which includes differentiating each term accordingly.
It helps to see that, in this equation, both x and y are treated implicitly as functions that change with each other. The next step involves another core concept – solving equations to isolate \(\frac{dy}{dx}\).

Solving equations is a fundamental skill in calculus and broader mathematics. After implicit differentiation, we obtain an equation involving \(\frac{dy}{dx}\). To find horizontal tangents, \(\frac{dy}{dx}\) must equal zero. To isolate \(\frac{dy}{dx}\), we first rearrange: \[8y \frac{dy}{dx} = -3x^{2} + 6x\].
By solving for \(\frac{dy}{dx}\), we get: \[ \frac{dy}{dx} = \frac{-3x^{2} + 6x}{8y} \]. For horizontal tangents, the derivative should be zero. Setting the numerator equal to zero gives us \[ -3x^{2} + 6x = 0 \], and we solve this to get x-values. This type of algebraic manipulation is common in calculus and helps us understand how changes occur across variables.

Calculus is all about understanding changes. In this specific problem, we're looking at how a curve behaves and finding specific points where certain conditions (horizontal tangents) occur. Through calculus techniques like differentiation, we analyze these behaviors.
Horizontal tangents are where the slope of the curve is zero. By taking implicit derivatives, rearranging, and solving, we find these unique points. This aids in a deeper comprehension of not just curves, but of how things change in immediate terms (derivatives), versus overall terms (integrals). Calculus connects these concepts to provide insights into various dynamic systems.

Curve analysis is crucial to understanding the graphical depiction of functions. Identifying points where the curve has horizontal tangents is a part of this analysis. The curve in this problem is given by \[x^{3} - 3x^{2} + 4y^{2} = 4\]. By performing the steps of calculus, we determine the points of interest.
We substitute the solved x-values back into the original curve equation to find corresponding y-values. For example, for x = 0, we solve \[0^{3} - 3(0)^{2} + 4y^{2} = 4\], to find y-values.
This gives rise to specific points, \( (0, 1) \, \ (0, -1) \, \ (2, \sqrt{2}) \, \ (2, - \sqrt{2}) \). Recognizing these points on the curve helps in understanding the shape and behavior of the function across its domain. This systematic approach is foundational in curve analysis, helping to uncover critical insights in graphical data.