Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = x³ / 3 - 9x

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To find the critical points of the function \( f(x) = \frac{x^3}{3} - 9x \), we first need to find its derivative, \( f'(x) \).

Differentiate \( f(x) \) with respect to \( x \). The derivative of \( \frac{x^3}{3} \) is \( x^2 \), and the derivative of \( -9x \) is \( -9 \). Therefore, \( f'(x) = x^2 - 9 \).

Set the derivative \( f'(x) = x^2 - 9 \) equal to zero to find the critical points: \( x^2 - 9 = 0 \).

Solve the equation \( x^2 - 9 = 0 \) for \( x \). This can be factored as \( (x - 3)(x + 3) = 0 \), giving the solutions \( x = 3 \) and \( x = -3 \).

The critical points of the function are \( x = 3 \) and \( x = -3 \). These are the values of \( x \) where the derivative is zero, indicating potential local maxima, minima, or points of inflection.

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주요 개념

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Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local maxima, minima, and points of inflection. To find critical points, one must first compute the derivative of the function and then solve for the values of x that satisfy the condition of the derivative being zero.

Derivative

The derivative of a function measures the rate at which the function's value changes with respect to changes in its input. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the function at any given point. For the function given, the derivative will help identify where the function's behavior changes, leading to the determination of critical points.

Local Extrema

Local extrema refer to the highest or lowest points in a particular interval of a function. These points are found at critical points where the derivative is zero or undefined. Understanding local extrema is crucial for analyzing the behavior of functions, as they indicate where the function reaches its maximum or minimum values within a specified range.

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.ƒ(x) = x³ / 3 - 9x