3. For what value of x is 42x+3 = 417 true?
the equation
=
4. Let the function f be defined as
f(x ) = 3x2 - 4(3x + 6). What is the
value of f(4) ?
=
-

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.