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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 4.5.67
Chapter 4, Problem 4.5.67

Theory and Examples
67. An inequality for positive integers Show that if a, b, c, and d are positive integers, then
[(a^2+1)(b^2+1)(c^2+1)(d^2+1)]/abcd ≥ 16

Verified step by step guidance
1
Recognize that the problem involves proving an inequality for positive integers a, b, c, and d. The expression to analyze is \( \frac{(a^2+1)(b^2+1)(c^2+1)(d^2+1)}{abcd} \geq 16 \).
Apply the AM-GM inequality, which states that for any non-negative numbers, the arithmetic mean is greater than or equal to the geometric mean. For two numbers x and y, this is \( \frac{x + y}{2} \geq \sqrt{xy} \).
Consider each term \( a^2 + 1 \), \( b^2 + 1 \), \( c^2 + 1 \), and \( d^2 + 1 \) separately. Apply the AM-GM inequality to each: \( a^2 + 1 \geq 2a \), \( b^2 + 1 \geq 2b \), \( c^2 + 1 \geq 2c \), and \( d^2 + 1 \geq 2d \).
Multiply the inequalities obtained from the AM-GM application: \( (a^2 + 1)(b^2 + 1)(c^2 + 1)(d^2 + 1) \geq (2a)(2b)(2c)(2d) = 16abcd \).
Divide both sides of the inequality by \( abcd \) to obtain \( \frac{(a^2+1)(b^2+1)(c^2+1)(d^2+1)}{abcd} \geq 16 \), thus proving the original inequality.

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Key Concepts

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

Inequality

An inequality is a mathematical statement that compares two values, expressions, or quantities, indicating that one is larger or smaller than the other. In this problem, the inequality involves positive integers and requires proving that a certain expression is greater than or equal to 16. Understanding inequalities is crucial for manipulating and comparing expressions to establish the required relationship.
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AM-GM Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) Inequality is a fundamental inequality in mathematics that states for any non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. This concept is often used to prove inequalities involving products and sums, and it can be applied here to show that the expression involving a, b, c, and d meets the required condition.
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Positive Integers

Positive integers are the set of all whole numbers greater than zero. In this problem, a, b, c, and d are specified as positive integers, which means they are natural numbers. This restriction is important because it influences the properties of the expressions involved, such as ensuring that the product abcd is non-zero, allowing for division and comparison in the inequality.
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Related Practice
Textbook Question

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.

39. y = 8 / (x² + 4) (Witch of Agnesi)

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.


∫(3t² + t/2) dt

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 = (𝓍 + 1) / (𝓍² + 2𝓍 + 2)

Textbook Question

Theory and Examples


Determine the values of constants a and b so that f(x) = ax² + bx has an absolute maximum at the point (1,2).

Textbook Question

30. Find a positive number for which the sum of its reciprocal and four times its square is the smallest possible.

Textbook Question

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

80. y' = 1 - cot²θ, for 0 < θ < π