Question

If f(x) = |(x^2 - 6)(x^2 + 2)|, how many numbers in the interval 1 ≤ x ≤ 2 satisfy the conclusion of the mean value theorem?

None
One
Two
Three

Not sure where to start. Like my previous question, I don't have a book or help from my teacher.

Answer 1

Ah, so you have heard the words "mean value theorem"

now let's see what the slope is from x = 1 to x = 2

y(2) = |(-2)(6)| = 12
y(1) = |(-5)(3)| = 15
so
slope = -3/1 = -3

Mean value theorem says slope must be 3 at least once between = 1 and x = 2

lets look at derivative (slope) of
y = x^4 - 4 x^2 - 12
dy/dx = 4 x^3 - 8 x

where is that equal to 3 between x = 1 and x = 2 ?
4 x^3 - 8 x = 3
graph that from x = 1 to x = 2
at 1 it is -4
at 2 it is 16
I bet you do not find it is 3 more than once :)
try this:
http://www.mathportal.org/calculators/polynomials-solvers/polynomial-graphing-calculator.php
you will find only once between 1 and 2 I think