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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.4.17
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

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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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Related Practice
Textbook Question

{Use of Tech} Newton’s method and curve sketching Use Newton’s method to find approximate answers to the following questions.

Where is the first local minimum of f(x) = (cos x)/x on the interval (0,∞) located?

Textbook Question

Approximating changes


Approximate the change in the lateral surface area (excluding the area of the base) of a right circular cone of fixed height h = 6m when its radius decreases from r = 10 m to r = 9.9 m (S = πr√(r² + h²).

Textbook Question

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.


ƒ(x) = 2x³ - 15x² + 24x on [0,5]

Textbook Question

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.


f(x) = sin x + x - 1; x₀ = 0.5

Textbook Question

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

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Textbook Question

Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.


v(t) = 6t² + 4t - 10; s(0) = 0

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