What is the result when 6x2 - 13x + 12 is subtracted from
-3x2 + 6x + 7?
(1) 3x2 - 7x + 19 (3) 9x2 - 7x + 19
(2) 9x2 - 19x + 5 (4) -9x2 + 19x - 5

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.