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Find the area of the following colored regions?

Area = 77.1

Step-by-step explanation:

The are of a right triangle is

our base and height are 6 for both triangles, so we shall find the area of one of the triangles...

multiply this by 2

Now find the area of the whole circle

our radius is 6, therefore:

Finally, subtract the area of the two triangles from the total are of the circle. The difference left over is our answer

The mean value theorem is used to link the average rate of change and the derivative of a function.

The minimum and the maximum possible values of f(6) - f(1) are 20 and 25, respectively

The given parameters are:

The mean value theorem states that:

If f(x) is continuous on [a,b] and differentiable on (a,b), then there exist a number c in (a,b) such that:

To find f(6) - f(1), we set the values of a and b to be:

So, we have:

The maximum value of f'(x) is 5.

So, we have:

Multiply both sides by 5

Rewrite as:

Similarly, the minimum value of f'(x) is 4.

So, we have:

Multiply both sides by 5

Rewrite as:

So, we have:

Hence, the minimum and the maximum possible values of f(6) - f(1) are 20 and 25, respectively

Read more about mean value theorems at:

link