Theory and Examples

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

Let f(x) = |x³ − 9x|.

b. Does f'(-3) exist?

검증된 단계별 안내

To determine if f'(-3) exists, we need to check if the function f(x) = |x³ − 9x| is differentiable at x = -3. Differentiability requires the function to be continuous and have a defined derivative at that point.

First, check the continuity of f(x) at x = -3. Since f(x) is an absolute value function, it is continuous everywhere, including at x = -3.

Next, consider the definition of the derivative. The derivative f'(x) exists at x = -3 if the limit of the difference quotient exists as x approaches -3 from both sides.

To find the derivative, consider the piecewise nature of the absolute value function. The expression inside the absolute value, x³ - 9x, changes sign at the roots of the equation x³ - 9x = 0. Solve for x to find these critical points.

Evaluate the derivative from the left and right of x = -3 using the piecewise definition of f(x). If the left-hand and right-hand derivatives are equal, then f'(-3) exists. Otherwise, it does not.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is the slope of the tangent line to the function's graph at that point. For a function f(x), the derivative is denoted as f'(x) and is found using differentiation rules.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x. It affects the differentiability of functions because it creates sharp corners or cusps in graphs, where the derivative may not exist. For f(x) = |x³ − 9x|, the absolute value impacts the function's smoothness and continuity.

Differentiability and Continuity

A function is differentiable at a point if it is smooth and has a defined tangent at that point, implying continuity. However, a function can be continuous but not differentiable at points where there are sharp turns or cusps, such as those introduced by absolute values. Checking differentiability involves examining the function's behavior around the point of interest.

Theory and ExamplesIn Exercises 51 and 52, give reasons for your answers.Let f(x) = |x³ − 9x|.b. Does f'(-3) exist?