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: f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]. We need to evaluate this limit at x = 3.

To evaluate the derivative, consider the piecewise nature of the absolute value function. For x³ - 9x, identify the intervals where the expression inside the absolute value is positive or negative. This will help in determining the left-hand and right-hand derivatives at x = 3.

Calculate the left-hand derivative (approaching from the left of x = 3) and the right-hand derivative (approaching from the right of x = 3). If both derivatives exist and are equal, then f'(3) exists. If they are not equal, f'(3) does not exist.

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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 f'(x) is found using limits, and it exists at a point if the function is continuous and smooth (differentiable) there.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x, regardless of its sign. When dealing with derivatives, the presence of an absolute value can create points where the function is not differentiable, typically at points where the expression inside the absolute value changes sign, leading to a cusp or corner in the graph.

Differentiability and Continuity

A function is differentiable at a point if it is smooth and has no sharp corners or cusps at that point, which also implies it must be continuous there. For f(x) = |x³ − 9x|, checking differentiability at x = 3 involves ensuring the function is continuous and that the left-hand and right-hand derivatives at x = 3 are equal.

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