Solve & plot a graph for the following: F(X)=cos(X)-cos to second power X for the interval negative pi greater than or equal to X less than or equal to positive pi

To solve and plot the graph of the function F(x) = cos(x) - cos^2(x) for the given interval [-π, π], we can follow these steps:

Step 1: Determine the domain and range of the function:
Since cosine is defined for all real numbers, the domain of F(x) is (-∞, ∞). The range of the function can be found by analyzing the minimum and maximum values of cos(x) and cos^2(x).

Step 2: Find the critical points:
To find the critical points, we need to determine where the derivative of F(x) is equal to zero. Let's first find the derivative of F(x):

F'(x) = -sin(x) - 2cos(x)(-sin(x)) = -sin(x) + 2sin(x)cos(x) = sin(x)(2cos(x) - 1)

Set F'(x) = 0 and solve the equation: sin(x)(2cos(x) - 1) = 0

For sin(x) = 0, x can be -π, 0, or π.

For 2cos(x) - 1 = 0, cos(x) = 1/2, x can be π/3 or 5π/3.

Therefore, the critical points are x = -π, π/3, 0, 5π/3, and π.

Step 3: Evaluate the endpoints:
Evaluate F(x) at both endpoints of the given interval: x = -π and x = π.

F(-π) = cos(-π) - cos^2(-π) = -1 - (1)^2 = -2
F(π) = cos(π) - cos^2(π) = -1 - (1)^2 = -2

Step 4: Plot the graph:
Now that we have the critical points, the endpoints, and their corresponding function values, we can plot the graph.

On the y-axis, plot the values -2.
On the x-axis, label the points -π, π/3, 0, 5π/3, π.

Connect the points with a smooth curve, making sure to capture the concave up and concave down shapes of the cosine function.

The resulting graph will show the behavior of the function F(x) = cos(x) - cos^2(x) over the interval [-π, π].