Graph each function and approximate any local maxima or minima to the nearest tenth.

1. P(x) = 2x^3 - 5x

2. P(x) = 2x^3 - 4x + 1

We can't draw graphs for you here. You will have to get some graph paper and draw your own. It will be good practice for you.

The first equation has a local maximum or minimum at x = + or - sqrt(5/6)
The second funtion has a local maximum or minimum at x = + or - sqrt(2/3)

When you draw the curves you will see which are maxima or minima. Both have one of each.

To graph each function and find the local maxima or minima, we need to follow these steps:

Step 1: Determine the derivative of the function.

The derivative of a function helps us find the critical points, which are potential locations of the local maxima or minima. For a function P(x), the derivative is denoted as P'(x), read as "P prime of x."

Step 2: Set the derivative equal to zero and solve for x.

This step helps us find the critical points. The points where the derivative is zero or undefined are the potential locations for the local maxima or minima.

Step 3: Determine the concavity of the function.

To identify if a critical point is a local maximum or minimum, we need to analyze the concavity. This can be done by finding the second derivative, denoted as P''(x) or the derivative of P'(x).

Step 4: Solve the second derivative.

Like in Step 2, we set the second derivative equal to zero to identify any inflection points, where the function changes concavity.

Step 5: Evaluate the function at the critical points.

Plug in the critical points from Step 2 into the original function to find the corresponding y-values.

Step 6: Plot the graph and label local maxima or minima.

Using the information from Steps 2, 3, 4, and 5, plot the graph of the function and indicate any local maxima or minima with their approximate values.

Now, let's apply these steps to each of the given functions:

1. P(x) = 2x^3 - 5x

Step 1: Find the derivative
P'(x) = 6x^2 - 5

Step 2: Set the derivative equal to zero and solve for x
6x^2 - 5 = 0
6x^2 = 5
x^2 = 5/6
x = +/- sqrt(5/6)

Step 3: Find the second derivative
P''(x) = 12x

Step 4: Solve the second derivative
12x = 0
x = 0

Step 5: Evaluate the function at critical points
P(0) = 0

Step 6: Plot the graph
The graph of P(x) = 2x^3 - 5x will be a cubic function. Plot the points (0,0) and consider concavity to label any local maxima or minima.

2. P(x) = 2x^3 - 4x + 1

Follow the same steps as above to find the critical points and concavity for P(x) = 2x^3 - 4x + 1.

Once you have plotted the graph of P(x) = 2x^3 - 4x + 1 and determined the local maxima or minima, label them accordingly with their approximate values rounded to the nearest tenth.