Domanda del libro di testo
Finding Extreme Values
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
y = 𝓍³ ― 2𝓍 + 4
206
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 4.4.53
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)
Guida verificata passo dopo passo1
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.
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:
18mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
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.
Video consigliato:
Percorso guidato
5:53
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.
Video consigliato:
04:50Critical 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.
Video consigliato:
04:50Critical Points
Pratica correlata
Domanda del libro di testo
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = √(x(1 − x)), [0, 1]
188
views
Domanda del libro di testo
Absolute Extrema on Finite Closed Intervals
In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.
f(x) = x⁴ᐟ³, −1 ≤ x ≤ 8
189
views
Domanda del libro di testo
Business and Economics
60. Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.
229
views
Domanda del libro di testo
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
g(t) = √t + √(1 + t) − 4, (0, ∞)
176
views
Domanda del libro di testo
Checking Antiderivative Formulas
Right, or wrong? Give a brief reason why.
∫−15(x + 3)² / (x − 2)⁴ dx = ((x + 3)/(x − 2))³ + C
28
views