Domanda del libro di testo
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).
212
views
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 4, Problema 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
Guida verificata passo dopo passo1
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.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
5mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
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.
Video consigliato:
Percorso guidato
5:10
Finding the Domain and Range of a Graph
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.
Video consigliato:
Percorso guidato
5:10Finding the Domain and Range of a Graph
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.
Video consigliato:
02:33Determining Concavity Given a Function Example 3
Pratica correlata
Domanda del libro di testo
30. Find a positive number for which the sum of its reciprocal and four times its square is the smallest possible.
191
views
Domanda del libro di testo
Applications
Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec². How fast will the rocket be going 1 min later?
19
views
Domanda del libro di testo
Finding Functions from Derivatives
Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.
205
views
Domanda del libro di testo
93. The accompanying figure shows a portion of the graph of a twice-differentiable function y=f(x). At each of the five labeled points, classify y' and \(\y\)'' as positive, negative, or zero.
" style="" width="254">
236
views
Domanda del libro di testo
32. Answer Exercise 31 if one piece is bent into a square and the other into a circle.
126
views