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)=2x^3-5x^2-x+1; [1.4, 2]

Verified step by step guidance

Step 1: Understand that turning points of a polynomial function occur where the derivative is zero or undefined. For the function \( f(x) = 2x^3 - 5x^2 - x + 1 \), we need to find its derivative.

Step 2: Differentiate the function \( f(x) \) to find \( f'(x) \). The derivative is \( f'(x) = 6x^2 - 10x - 1 \).

Step 3: Set the derivative \( f'(x) = 6x^2 - 10x - 1 \) equal to zero to find the critical points: \( 6x^2 - 10x - 1 = 0 \).

Step 4: Solve the quadratic equation \( 6x^2 - 10x - 1 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 6 \), \( b = -10 \), and \( c = -1 \).

Step 5: Evaluate the solutions of the quadratic equation within the given domain \([1.4, 2]\) to find the coordinates of the turning points. Use a graphing calculator to determine the \( y \)-values for these \( x \)-coordinates and round to the nearest hundredth.

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Key Concepts

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

Turning Points

Turning points are points on a graph where the function changes direction from increasing to decreasing or vice versa. For polynomial functions, these points occur where the first derivative of the function equals zero. Identifying turning points is crucial for understanding the shape and behavior of the graph.

Derivatives

The derivative of a function measures the rate at which the function's value changes as its input changes. In the context of polynomial functions, the first derivative is used to find critical points, which are potential turning points. Calculating the derivative is essential for analyzing the function's behavior over a specified interval.

Graphing Calculators

Graphing calculators are powerful tools that allow users to visualize functions and perform complex calculations, including finding roots and derivatives. They can plot graphs, compute turning points, and provide numerical approximations. Familiarity with using a graphing calculator is important for efficiently solving problems involving polynomial functions.