Im a bit stuck on this question. im fine when i need to simplify, but i guess its when i combine terms where i go wrong. may i get some ?

Multiply everything out first.
12x^3+8x^2+4x+6x^2+4x+2

Reorganize to make this easier.
(12x^3)+(8x^2+6x^2)+(4x+4x)+(2)

Combine like terms.
12x^3+14x^2+8x+2

Hope this helps!

11554567788908766449776544

Answer

1 number

explanation

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.