91Ó°ÊÓ

Critical points are an essential concept in calculus. These points occur in a function where the first derivative, \( f'(x) \), is zero or undefined. In simple terms, they are where the slope of the tangent to the curve is flat, indicating potential maxima, minima, or points of inflection.

To find critical points, differentiate the function and set this derivative equal to zero. For example, given \( f(x) = x^3 - 3x^2 \), the first derivative is \( 3x^2 - 6x \). Setting \( f'(x) = 0 \), solve for \( x \):

• \( 3x(x-2) = 0 \)
• Thus, \( x = 0 \) or \( x = 2 \)

These points are where the function changes direction and are critical for understanding the function's behavior between defined intervals. In the interval \([-1, 3]\), \( x = 0 \) and \( x = 2 \) are crucial for further analysis.

Inflection points represent locations on the curve where the concavity changes. A function changes from concave up (shaped like a cup) to concave down (shaped like a cap), or vice versa, at these points.

To determine inflection points, find where the second derivative, \( f''(x) \), equals zero or changes sign. For our example function, \( f''(x) = 6x - 6 \). Setting \( f''(x) = 0 \) leads us to:

• \( 6x - 6 = 0 \)
• So, \( x = 1 \)

Next, check the sign of \( f''(x) \) on either side of \( x = 1 \):

• If \( x < 1 \), \( f''(x) < 0 \) (concave down)
• If \( x > 1 \), \( f''(x) > 0 \) (concave up)

This confirms \( x = 1 \) as an inflection point, showing the point where the curvature of the graph changes.

Extrema refers to the highest and lowest points in a specific part of a function's graph. Understanding local and global extrema helps to identify peaks and troughs of a function.

**Local Extrema:** These are the highest or lowest points within a small range of the function. The function \( f(x) = x^3 - 3x^2 \) has critical points at \( x = 0 \) and \( x = 2 \), which can be examined for local maxima or minima.

**Global Extrema:** These represent the absolute highest and lowest values over the function's entire domain. It is crucial to evaluate the function not only at critical points but also at the boundaries of the domain, \(-1\) and \(3\):

• At \( x = -1 \), \( f(-1) = -4 \)
• At \( x = 0 \), \( f(0) = 0 \)
• At \( x = 2 \), \( f(2) = -4 \)
• At \( x = 3 \), \( f(3) = 0 \)

In this case, local minima occur at \((-1, -4)\) and \((2, -4)\), while local maxima are at both \((0, 0)\) and \((3, 0)\). The global minimum value is \(-4\), as this is the lowest value across the entire interval. There are no global maxima as higher values don’t exist outside the local context in this domain.