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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.4.53
Chapter 4, Problem 4.4.53

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
53. y = x * √(8 - x²)

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First, identify the domain of the function y = x * √(8 - x²). The expression under the square root, 8 - x², must be non-negative. Solve the inequality 8 - x² ≥ 0 to find the domain.
Next, find the first derivative of the function to identify critical points. Use the product rule and chain rule to differentiate y = x * √(8 - x²). Set the derivative equal to zero to find critical points, which may correspond to local extreme points.
Determine the second derivative to find inflection points. Differentiate the first derivative to obtain the second derivative. Set the second derivative equal to zero and solve for x to find potential inflection points.
Evaluate the function at the critical points and endpoints of the domain to find local extreme points. Compare these values to determine if they are local maxima or minima.
Finally, evaluate the function at the critical points and endpoints of the domain to find absolute extreme points. Compare these values to determine the absolute maximum and minimum values of the function within the domain.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Functions

Graphing functions involves plotting the curve of a function on a coordinate plane to visualize its behavior. This includes identifying key features such as intercepts, asymptotes, and the general shape of the graph. Understanding how to graph functions is essential for analyzing their properties and behavior over different intervals.
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Graph of Sine and Cosine Function

Local Extreme Points

Local extreme points are points on the graph where the function reaches a local maximum or minimum. These occur where the derivative of the function equals zero or is undefined, indicating a change in direction. Identifying these points helps in understanding the function's behavior in specific regions and is crucial for optimization problems.
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Critical Points

Inflection Points

Inflection points are points on the graph where the curvature changes direction, from concave up to concave down or vice versa. These are found by analyzing the second derivative of the function. Inflection points provide insight into the function's concavity and are important for understanding the overall shape and behavior of the graph.
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Critical Points
Related Practice
Textbook Question

Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


y = x² + 2/x

Textbook Question

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

______

y = √𝓍² ― 1

Textbook Question

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


d²r/dt² = 2/t³; dr/dt|ₜ ₌ ₁ =1, r(1) = 1

Textbook Question

113. If b, c, and d are constants, for what value of b will the curve y = x^3 + bx^2 + cx + d have a

point of inflection at x = 1? Give reasons for your answer.

Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(−2cost) dt

Textbook Question

37. What value of a makes f(x) = x^2 +(a/x) have

a. a local minimum at x = 2?

b. a point of inflection at x = 1?