Ah cont'd

Putting numbers into the average value theorem:

For y = (2/3)*x
The integral:

Thus, setting C = 0

100/3 - 4/3 = 96/3
                   = 32

Applying the theorem,

v = 1/(10 - 2) * 32
   = 4

 The average value is 4.

True, by just looking at our function at its midpoint on the x-axis, 6

(2/3)*6 = 4

Not a proof, but certainly a demonstration

                                                        *     *     *

    4 = (x^2)/3

x^2 = 12

       = 4 * 3

    x = ± 2*3^.5

       = 3.464

   

                                                       *     *     *

I thought at first that complex numbers might facilitate life
with square, cubic...n roots. After all, a complex number always
has at least one root: that is the Fundamental Theorem of Algebra
(which made Gauss famous). Turns out nope, because these roots
are still dangerous. Consider:

(-2)^3 = -8 so that
(-8)^1/3 = -2, is its only root. Fine and good.

But if one starts to work with this, one might find that

source: German-laguage Wikipedia

In terms of current work in mathematics, complex roots

are studied on Riemann surfaces (named in honor of

Riemann).