Use a graphics utility to plot the following curves. Where...

Use a graphics utility to plot the following curves. Where does the curve have a horizontal tangent? Does the curve fail to have a tangent line anywhere?
$$y=\left(\left(x^{2}-1\right)^{2}\right)^{1 / 3}$$

Key Concepts

Differentiation Techniques (Chain Rule and Power Rule)

Differentiating a composite function, especially one involving a fractional exponent or nested expressions, requires the use of the chain rule combined with the power rule. The chain rule is applied when a function is composed of another function, allowing for systematic computation of the derivative. Mastery of these techniques is essential in finding expressions for the slope of the tangent line and analyzing critical points of the function.

Horizontal Tangents

A horizontal tangent occurs at a point on a curve where the derivative (the slope of the tangent line) is zero. In the context of calculus, identifying horizontal tangents helps to locate potential local extrema or points of inflection. The process typically involves differentiating the function and setting the derivative equal to zero, then solving for the corresponding x-values, which, when substituted back into the function, provide the exact points on the curve.

Non?Differentiability (Cusps and Corners)

A function may fail to have a tangent line at certain points if it is non?differentiable there. This typically happens at cusps or corners where the left-hand and right-hand derivatives do not coincide, or where the derivative becomes infinite. Analyzing these points is crucial to understanding the overall behavior of a function, as they indicate sudden changes in the curve’s direction that cannot be captured by a single tangent line.

Graphical Analysis in Calculus

Graphical analysis provides a visual interpretation of a function’s behavior, including where horizontal tangents occur and points of non-differentiability. Using technology and graphing utilities, one can plot the function to observe its shape, confirm analytical findings, and gain insights into properties such as continuity, smoothness, and the existence of cusps or vertical tangents.

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