Explanation:

To calculate the Y values corresponding to each X value, we can substitute the given X values into the given function

Y = X^3 - 5X^2 - 8X + 20.

Here are the calculations:

For X = -1.333:

Y = (-1.333)^3 - 5(-1.333)^2 - 8(-1.333) + 20 Y

= -3.555 - 11.109 + 10.664 + 20 Y

= 16.999

For X

= -0.667: Y

= (-0.667)^3 - 5(-0.667)^2 - 8(-0.667) + 20 Y

= -0.257 - 1.852 + 5.336 + 20 Y

= 23.227

For X = 1.667:

Y = (1.667)^3 - 5(1.667)^2 - 8(1.667) + 20 Y

= 4.267 - 13.889 - 13.336 + 20 Y

= -2.958

The calculated Y values and their signs are as follows:

X | Y value | Sign

-1.333 | 16.999 | + -0.667 | 23.227 | + 1.667 | -2.958 | -

To find the critical values among the given X values, we need to find the first and second derivatives of the given function.

1. First derivative (dy/dx): To find the first derivative of the function Y = X^3 - 5X^2 - 8X + 20, we differentiate each term with respect to X:

dy/dx

= d/dx (X^3) - d/dx (5X^2) - d/dx (8X) + d/dx (20)

= 3X^2 - 10X - 8

2a. Solve the first derivative:

To find the X values that make the first derivative equal to zero

(dy/dx = 0), we set

dy/dx = 0 and solve for X:

3X^2 - 10X - 8 = 0

We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.

3b. Critical values:

The X values where dy/dx = 0 are the critical values. If any such X values are found, we mark them with "O" in the dy/dx row. However, we need to solve the quadratic equation from step 2a to find the specific values of X.

1. Second derivative (d'y/dx?): To find the second derivative of the function Y = X^3 - 5X^2 - 8X + 20, we differentiate the first derivative (dy/dx) with respect to X:

d'y/dx?

= d/dx (3X^2 - 10X - 8)

= 6X - 10

4a. Solve the second derivative:

To find the X values that make the second derivative equal to zero (d'y/dx? = 0), we set d'y/dx? = 0 and solve for X:

6X - 10 = 0

4b. Critical values:

The X value where d'y/dx? = 0 is a critical value. If such an X value is found, we mark it with "O" in the d'y/dx? row. Again, we need to solve the equation from step 4a to find the specific value of X.

To determine the maximums, minimums, and inflection points among the X,Y pairs, we need to analyze the critical values we found earlier.

1. Maximums: For a maximum, both dy/dx = 0 and dy/dx? < 0. Let's refer to the X values and their corresponding dy/dx? values below:

X value | dy/dx?

There are no maximums among the given X,Y pairs since there are no X values where dy/dx? < 0.

1. Minimums: For a minimum, both dy/dx = 0 and dy/dx? > 0. Let's refer to the X values and their corresponding dy/dx values below:

X value | dy/dx

There are no minimums among the given X,Y pairs since there are no X values where dy/dx > 0.

1. Inflection points: For an inflection point, dy/dx may or may not be equal to zero, but dy/dx? = 0. Let's refer to the X values and their corresponding dy/dx? values below:

X value | dy/dx?

There is one inflection point among the given X,Y pairs, where dy/dx? = 0. The X value for this inflection point is -1.333, and its dy/dx? value is 0.

Bonus: To plot the graph with the X, Y pairs, you can use a software like Excel and label the maximums, minimums, and inflection points on the graph, as well as their respective X and Y values. Scale the axes to clearly display these points.