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, 0]

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Identify the polynomial function: \( f(x) = 2x^3 - 5x^2 - x + 1 \).

Determine the derivative of the function to find the critical points: \( f'(x) = 6x^2 - 10x - 1 \).

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

Solve the quadratic equation for \( x \) to find the critical points within the interval \([-1, 0]\).

Use a graphing calculator to evaluate \( f(x) \) at the critical points and endpoints to find the turning points, rounding 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 at a given point. For polynomial functions, the first derivative can be calculated using power rules. Finding the derivative is essential for determining the turning points, as these occur where the derivative is zero.

Graphing Calculators

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