Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
g(x) = √(2x − x²)
5. Graphical Applications of Derivatives
The First Derivative Test
4:11 minutesProblem 4.1.52c
Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
b. Does f'(-3) exist?
Verified step by step guidance1
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.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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