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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 4.4.26
Chapter 4, Problem 4.4.26

Graphing functions Use the guidelines of this section to make a complete graph of f.


f(x) = x³ - 147x + 286

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Identify the critical points of the function by finding the derivative of f(x) = x³ - 147x + 286. Set the derivative equal to zero to find the critical points.
Calculate the derivative: f'(x) = 3x² - 147. Set this equal to zero and solve for x to find the critical points: 3x² - 147 = 0.
Solve the equation 3x² - 147 = 0 for x. This will give you the x-values of the critical points.
Determine the nature of each critical point by using the second derivative test. Calculate the second derivative: f''(x) = 6x, and evaluate it at each critical point.
Analyze the behavior of the function as x approaches positive and negative infinity to understand the end behavior of the graph. This will help in sketching the complete graph of the function.

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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. The function f(x) = x³ - 147x + 286 is a cubic polynomial, characterized by its highest degree of 3. Understanding the general shape and behavior of polynomial functions is essential for graphing them accurately.
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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 and turning behavior of the graph.
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End Behavior of Functions

End behavior describes how a function behaves as the input values approach positive or negative infinity. For cubic functions like f(x), the end behavior is determined by the leading term, which in this case is x³. As x approaches infinity, f(x) also approaches infinity, and as x approaches negative infinity, f(x) approaches negative infinity, shaping the overall graph.
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