Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-(1/3)x⁻⁴ᐟ³
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 4.3.1c
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = x(x − 1)
Guida verificata passo dopo passo1
To find the points where the function f assumes local maximum or minimum values, we need to identify the critical points. Critical points occur where the derivative f'(x) is zero or undefined. Start by setting the derivative equal to zero: f'(x) = x(x - 1) = 0.
Solve the equation x(x - 1) = 0 to find the critical points. This equation can be factored into two parts: x = 0 and x - 1 = 0. Solving these gives the critical points x = 0 and x = 1.
Next, determine whether each critical point is a local maximum, minimum, or neither by using the first derivative test. This involves analyzing the sign of f'(x) around each critical point.
Choose test points in the intervals determined by the critical points: for example, pick a point less than 0, between 0 and 1, and greater than 1. Evaluate the sign of f'(x) at these test points to determine the behavior of f(x).
If f'(x) changes from positive to negative at a critical point, f has a local maximum there. If f'(x) changes from negative to positive, f has a local minimum. If there is no sign change, the critical point is neither a maximum nor a minimum.
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Critical Points
Critical points of a function occur where its derivative is zero or undefined. These points are potential locations for local maxima or minima. To find them, set the derivative equal to zero and solve for x. In this case, solve f′(x) = x(x − 1) = 0 to find the critical points.
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04:50
Critical Points
First Derivative Test
The First Derivative Test helps determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign changes of the derivative around the critical points, one can infer the behavior of the function. If f′ changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it's a local minimum.
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07:09The First Derivative Test: Finding Local Extrema
Local Maximum and Minimum
A function has a local maximum at a point if the function value at that point is greater than at nearby points. Conversely, a local minimum occurs if the function value is less than at nearby points. Identifying these points involves analyzing the critical points and using tests like the First Derivative Test to confirm the nature of each point.
Video consigliato:
06:02The Second Derivative Test: Finding Local Extrema
Pratica correlata
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