We draw the graph of f(x) = x²+2.

We find f(0) = 0²+2 = 2 and f(2) = 2²+2 = 6

Then the points (0,f(0)) and (2,f(2)) are (0,2) and (2,6),

We plot those and draw a line through them
(the green line below): 



Next we find the slope of that green line, using the
slope formula from algebra:

m = 
where (x1,y1) = (0,2)
and where (x2,y2) = (2,6)

m =  =  = 2

Now we want to find a point on the graph, between those
two points where the slope of a tangent line is parallel
to the green line.

The derivative is a formula for the slope of the tangent
line at any point we substitute the x-coordinate in.

f(x) = x²+2
f'(x) = 2x

We set 2x equal to the slope 2 of that green line.

2x = 2
 x = 1

That means that at the point (1,f(1)) which is (1,1²+2)
or (1,3), if we draw a line tangent to the graph at that
point it will be parallel to the green line.  So we draw
a tangent line at the point (1,3) (in blue), and we notice
that it is parallel to the green line:



I have drawn a black line from the point of tangency
of the blue line down to the x-axis to show that the value of 
c on the x-axis in the interval (0,2) is the value c=1.

What this is all about is showing you that if you have a
line that cuts a curve in two points there is always a point 
between them where you can draw a tangent line paralell to
that line that cuts through the curve.  Then the x-coordinate
of that point of tangency is the value of c.

Edwin