Lagrange's mean value Theorem

If f is continuous in [a,b] and derivable in (a,b), then

Demonstration: the equation of the secant line is

If d(x) is the difference between graph and secant

d is continuous in [a,b] and derivable in (a,b), and  d(a) = d(b) = 0, then (Rolle’s Theorem)

Exercise:

(reserva1- PAEG 2012)

Demonstrate that there exists a point in the interval [0,2] in which the tangent line to the graph of f(x) = 3x² +2x+1 is parallel to the segment that joins the points of the graph in  x = 0 and x = 2 and calculate it.