Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = xe⁻(ˣ²/₂)
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.4.17
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x³ - 6x² + 9x
Verified step by step guidance1
Step 1: Identify the domain of the function f(x) = x³ - 6x² + 9x. Since this is a polynomial function, the domain is all real numbers.
Step 2: Find the critical points by taking the derivative of f(x) to find f'(x). The derivative is f'(x) = 3x² - 12x + 9. Set f'(x) = 0 to find the critical points.
Step 3: Solve the equation 3x² - 12x + 9 = 0 to find the values of x where the function has critical points. Use the quadratic formula or factorization to find these values.
Step 4: Determine the nature of each critical point by using the second derivative test. Find f''(x) by differentiating f'(x), which gives f''(x) = 6x - 12. Evaluate f''(x) at each critical point to determine if they are local minima, maxima, or points of inflection.
Step 5: Analyze the behavior of the function as x approaches positive and negative infinity to understand the end behavior. Since the leading term is x³, the function will tend to infinity as x approaches infinity and negative infinity as x approaches negative infinity. Use this information along with the critical points to sketch the graph.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Functions
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, f(x) = x³ - 6x² + 9x is a cubic polynomial, which means its highest degree is three. Understanding the general shape and behavior of polynomial functions is crucial for graphing them accurately.
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Introduction to Polynomial Functions
Critical Points and Extrema
Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. To find these points for f(x), we need to compute its derivative, set it to zero, and solve for x. Analyzing these points helps in determining the overall shape of the graph.
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End Behavior
End behavior describes how a function behaves as the input values approach positive or negative infinity. For polynomial functions, the leading term dictates this behavior. In the case of f(x), since the leading term is x³, the graph will rise to positive infinity as x approaches positive infinity and fall to negative infinity as x approaches negative infinity, shaping the overall graph.
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03:07Cases Where Limits Do Not Exist
Related Practice
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→2π (x sin x + x² - 4π²) / (x - 2π)
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Textbook Question
Use linear approximation to estimate f (5.1) given that f(5) = 10 and f'(5) = -2.
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Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_y→2 (y²+y-6) / (√(8-y²)-y)
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Textbook Question
{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = 2x cos x on [0,2]
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Textbook Question
Use linear approximation to estimate f (3.85) given that f(4) = 3 and f'(4) = 2.
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