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One of the fundamental applications of the first derivative in calculus is determining where a function is increasing or decreasing. This is helpful for understanding the overall behavior or trend of the function. For the function given, \(f(x) = x^{2/3} - x^{1/3}\), the process starts by finding the first derivative using the power rule, resulting in \(f'(x) = \frac{2}{3}x^{-1/3} - \frac{1}{3}x^{-2/3}\).
Critical points occur where this derivative is zero or does not exist. Solving for \(f'(x) = 0\) identifies the critical point at \(x = 0\).
The first derivative is undefined at \(x=0\) also, creating another critical consideration.

To determine where the function increases or decreases, test the sign of the derivative on either side of these critical points:

• For \(x < 0\), the derivative \(f'(x)\) is positive, indicating the function is increasing.
• For \(x > 0\), the derivative \(f'(x)\) is negative, indicating the function is decreasing.

Therefore, \(f(x)\) is increasing on the interval \(-\infty < x < 0\) and decreasing on \(0 < x < \infty\). Observing these trends can help understand how the function behaves as \(x\) moves through its domain.

Concavity relates to how a function bends or curves, and it's evaluated using the second derivative of the function. When the second derivative is positive, the function is 'concave up,' resembling a cup opening upwards. When negative, it is 'concave down,' much like an upside-down cup.
For our function \(f(x) = x^{2/3} - x^{1/3}\), we begin by finding the second derivative: \(f''(x) = -\frac{2}{9}x^{-4/3} + \frac{2}{9}x^{-5/3}\).

Determine the intervals of concavity by assessing the sign of the second derivative for \(x < 0\) and \(x > 0\):

• For both \(x > 0\) and \(x < 0\), \(f''(x)\) remains positive, indicating that the function is concave up on both sides.

This means the graph of the function curves upwards throughout its domain. Points of inflection, which occur where concavity changes, would typically be marked by \(f''(x) = 0\), but here \(f''(x)\) is never zero. Thus, there are no points of inflection between intervals. Observing the concavity on a graph illustrates how the curve nature of \(f(x)\) may affect its increasing or decreasing tendencies.

Examining vertical tangents or cusps is crucial for understanding any abrupt changes or undefined behavior in the function's graph. Typically, a vertical tangent occurs when the slope (first derivative) becomes infinite at a point, while a cusp is formed when there's a sharp corner on the graph.

For the function in question, we explore \(f'(x)\), which is \(\frac{2}{3}x^{-1/3} - \frac{1}{3}x^{-2/3}\).
Notice that at \(x = 0\), \(f'(x)\) is not defined since the terms involve negative exponents, which might suggest potential for vertical behavior.

However, further analysis shows that the first derivative does not approach infinity around \(x = 0\). As such, neither vertical tangents nor cusps are present in this function. Hence, \(f(x)\) does not exhibit any of the sudden shifts typically indicated by these features. Confirming with a graph can ensure these sudden behaviors are accurately described by the function's derivative properties.