Using the method of completing the square, put each circle into the form

then determine the center and radius of each circle

Standard form:

Center:

Radius:

Step-by-step explanation:

The equation of a circle in the standard form is

Where the point (h, k) is the center of the circle

To transform this equation this equation  in the standard form we use the method of square.

First, we group similar variables

Divide both sides of equality by 4

Now we complete square for variable x.

Take the coefficient "b" that accompanies the variable x and divide by 2. Then, elevate the result to the square:

Now add on both sides of the equality

Factor the expression and simplify the independent terms

Then

and the center is

radius

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.