Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

d. Determine all extrema of f.

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To find the extrema of the function \( f(x) = |x^3 - 9x| \), we first need to consider the critical points of the function. Critical points occur where the derivative is zero or undefined.

First, find the derivative of the inside function \( g(x) = x^3 - 9x \). The derivative is \( g'(x) = 3x^2 - 9 \).

Set \( g'(x) = 0 \) to find the critical points of \( g(x) \). Solve \( 3x^2 - 9 = 0 \) to find the values of \( x \) where the derivative is zero.

The absolute value function \( f(x) = |g(x)| \) can change its behavior at points where \( g(x) = 0 \). Solve \( x^3 - 9x = 0 \) to find these points.

Evaluate \( f(x) \) at the critical points and endpoints of the domain to determine the extrema. Consider the behavior of \( f(x) \) around these points to classify them as local minima, maxima, or neither.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Absolute Value Function

The absolute value function, denoted as |x|, represents the distance of a number x from zero on the number line, always yielding a non-negative result. In the context of f(x) = |x³ − 9x|, it affects how we find extrema by considering both the positive and negative scenarios of the expression inside the absolute value.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are potential candidates for local extrema. For f(x) = |x³ − 9x|, we first need to find the derivative, set it to zero, and solve for x to identify critical points, considering the behavior of the absolute value function.

Extrema

Extrema refer to the maximum and minimum values of a function. To determine extrema, evaluate the function at critical points and endpoints of the domain. For f(x) = |x³ − 9x|, analyze the critical points and the behavior of the function as x approaches positive and negative infinity to identify all local and global extrema.

Theory and ExamplesIn Exercises 51 and 52, give reasons for your answers.Let f(x) = |x³ − 9x|.d. Determine all extrema of f.