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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 85
Chapter 4, Problem 85

Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial function in the given domain interval. Give answers to the nearest hundredth. ƒ(x)=2x3-5x2-x+1; [1.4, 2]

Verified step by step guidance
1
Understand that turning points of a polynomial function occur where the derivative equals zero, as these points correspond to local maxima or minima on the graph.
Find the first derivative of the function \( f(x) = 2x^3 - 5x^2 - x + 1 \). Use the power rule for differentiation: \( f'(x) = 6x^2 - 10x - 1 \).
Use the graphing calculator to graph the derivative function \( f'(x) = 6x^2 - 10x - 1 \) over the domain interval \([1.4, 2]\) and find the values of \( x \) where \( f'(x) = 0 \). These \( x \)-values are the candidates for turning points within the interval.
For each \( x \)-value found, substitute it back into the original function \( f(x) = 2x^3 - 5x^2 - x + 1 \) to find the corresponding \( y \)-coordinate of the turning point.
Round the coordinates \( (x, f(x)) \) of each turning point to the nearest hundredth as required.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Turning Points of a Polynomial Function

Turning points are points on the graph where the function changes direction from increasing to decreasing or vice versa. For polynomial functions, these correspond to local maxima or minima, which occur where the derivative equals zero. Identifying turning points helps understand the shape and behavior of the graph.
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Maximum Turning Points of a Polynomial Function

Using a Graphing Calculator to Find Turning Points

A graphing calculator can approximate turning points by graphing the function and using built-in tools like 'maximum' and 'minimum' within a specified domain. This method provides numerical coordinates of turning points, especially useful when algebraic solutions are complex or not easily obtainable.
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Maximum Turning Points of a Polynomial Function

Domain Restriction and Rounding

Restricting the domain to [1.4, 2] means only turning points within this interval are considered. Rounding answers to the nearest hundredth ensures numerical results are precise but manageable, which is important for practical interpretation and reporting of coordinates.
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Related Practice
Textbook Question

Solve each inequality. Give the solution set in interval notation. 4/(x+6)>2/(x-1)

Textbook Question

Graph each rational function. ƒ(x)=(20+6x-2x2)/(8+6x-2x2)

Textbook Question

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. ƒ(x)=−8x4+3x3−6x2+5x−7ƒ(x)=-8x^4+3x^3-6x^2+5x-7

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Textbook Question

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. ƒ(x)=6x4+2x3+9x2+x+5ƒ(x)=6x^4+2x^3+9x^2+x+5

Textbook Question

The distance between the two points P(x1,y1)P(x_1,y_1) and R(x2,y2)R\(\left\)(x_2,y_2\(\right\)) is d(P,R)=(x1−x2)2+(y1−y2)2d(P, R) = \(\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\). Distance formula. Find the closest point on the line y=2xy = 2x to the point (1,7)(1, 7). (Hint: Every point on y=2xy = 2x has the form (x,2x)(x, 2x), and the closest point has the minimum distance.)

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Textbook Question

A quadratic equation ƒ(x) = 0 has a solution x = 2. Its graph has vertex (5, 3). What is the other solution of the equation?