Consider the graph of the line y= 1/2x-4 and the point (-4,2)

the perimeter is increased by a factor of 2

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.

answer: translate the circles so they share a common center point and dilate circle y by a scale factor of 3.