Question

Use a table to graph f(x)=2x3−4x2−3x+4

on a separate sheet of paper. Then estimate the x-coordinates at which relative maxima and relative minima occur. Round your answer to the nearest tenth.

The relative maximum occurs between the integers x =

and x =
;

the relative minimum occurs between the integers x =

and x =

.

Answer 1

The relative maximum occurs between the integers x = -1.3 and x = -1.2; the relative minimum occurs between the integers x = 0.3 and x = 0.4.

To estimate the x-coordinates at which relative maxima and relative minima occur, you can follow these steps:

1. Create a Table: Generate a table of x and f(x) values. Choose x values around the critical points where the function might have maxima or minima. For this equation, consider values close to -1 and 0.

`x & \quad f(x) \\ -1.5 & \quad \\ -1.4 & \quad \\ -1.3 & \quad \\ -1.2 & \quad \\ -1.1 & \quad \\ -1.0 & \quad \\ 0.0 & \quad \\ 0.1 & \quad \\ 0.2 & \quad \\ 0.3 & \quad \\ 0.4 & \quad \\ 0.5 & \quad \\`

2. Evaluate f(x): Substitute each x value into the function \(f(x) = 2x^3 - 4x^2 - 3x + 4\) and calculate the corresponding f(x) values.

3. Identify Turning Points: Look for changes in the trend of the function. A relative maximum occurs where the function changes from increasing to decreasing, and a relative minimum occurs where the function changes from decreasing to increasing.

4. Estimate Intervals: Based on the table, estimate the intervals where these turning points occur. Round the x-coordinates to the nearest tenth.

By following these steps, you can estimate that the relative maximum occurs between x = -1.3 and x = -1.2, and the relative minimum occurs between x = 0.3 and x = 0.4.