factor out the gcf of three terms, then complete the factorization of 6x^3 + 6x^2 - 120x
possible answers are:
6x^2(x+1)-120x
6x(x+4)(x-5)
6x(x-4(x+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.

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