We observe the leading term -2x6. The degree is the exponent
6, an even number. That tells us that the extreme left and right behaviors
are the same. The negative coefficient -2 tells us that both extreme
left and right behaviors are downward. Thus the function has a maximum
value.
What is the value? What is the corresponding value of x?
We can only answer those in reverse order. So we answer first:
"At what value of x does this maximum value occur?"
We find the derivative
Set y' = 0
We divide thru by the constant -12
Factor the sum of two cubes:
The real roots are x=0, x=-2 (the third factor yields only complex roots.)
We do a first derivative test of x=0
intervals | x < -2 | -2 < x < 0 | x > 0
----------------------------------------------
test value t | -3 | 1 | 3
sign of y'(t) | + | - | -
direction | upward | downward | downward
(incr.) (decr.) (decr.)
At -2 there is a change from increasing (upward) to decreasing (downward)
Therefore there is a relative maximum pt. at x = -2.
At 0 there is no change from decreasing (downward), thus a horizontal
inflection point at x=0.
There are no other critical values of the derivative, so
the relative maximum pt. at x = -2, is an absolute maximum point.
Now we answer this question:
What is the maximum value reached at x = -2.
We substitute in the original equation:
So the maximum value is 10. Observe the maximum point (-2,10),
and also the horizontal inflection point at (0,-118)
Edwin