91Ó°ÊÓ

In calculus, identifying critical points is crucial for understanding the behavior of a function. A critical point for a function occurs where its derivative is either zero or undefined. These points are significant because they often indicate where a function changes its increasing or decreasing nature, potentially marking spots for maximum or minimum values.
To find the critical points for a function like \(y = 2x - 3x^{2/3}\), we start by calculating the first derivative:

• First derivative: \(\frac{dy}{dx} = 2 - 2x^{-1/3}\).

This derivative defines how the function changes locally. Setting the derivative to zero finds values of \(x\) where the function might change direction:

• Solve \(2 - 2x^{-1/3} = 0\) to get \(x = 1\).
• Identify undefined points, noticed here when \(x = 0\).

Hence, the critical points for the function are \(x = 1\) and \(x = 0\). These critical points prompt further examination to find out if they are maxima or minima of the function.

Extrema, in calculus, refer to points where a function reaches minimum or maximum values. By analyzing subtle changes in a function, we can identify its local and absolute extrema.
With our function \(y = 2x - 3x^{2/3}\), the critical points found at \(x = 1\) and \(x = 0\) lead to a deeper look into the nature of these points through the second derivative test.

• Compute the second derivative: \(\frac{d^2y}{dx^2} = -\frac{2}{3}x^{-4/3}\).
• Test these values, especially at \(x = 1\): \(\frac{d^2y}{dx^2} \bigg|_{x = 1} = -\frac{2}{3}\), which indicates a local maximum since it is negative.
• At \(x = 0\), the derivative's undefined behavior suggests neither a maximum nor a minimum.

Thus, \(x = 1\) is our local maximum, specifically at the point \((1, -1)\) with the functional value \(y(1) = -1\).
This is a local extremum, providing valuable information about the highest value the function attains in this specific region of its domain.

An inflection point in calculus represents where a function changes its curvature, from concave up to concave down, or vice versa. These points are where the second derivative changes sign.
In our exploration of the function \(y = 2x - 3x^{2/3}\), we focus on the second derivative \(\frac{d^2y}{dx^2} = -\frac{2}{3}x^{-4/3}\) to find these particular points:

• An inflection point occurs at \(x = 0\), where the second derivative transitions in sign.
• It's essential to examine changes closely around this point, ensuring that the transition from positive to negative (or vice versa) holds true.

This transition confirms \(x = 0\) as a point of inflection.
Understanding inflection points helps provide insight into the nuances of the curve's shape and its direction changes.