Differentiation

You're right - you can't use sub/super scripts - so we tend to use the notation x^2 or x^3 (as found on some calculators)

Ok so if you differentiate the above function then you get

12x^2 + 18x - 12

When that function = 0, you have a min or a max point

Since it is equal to zero you can divide throughout by 6 to get:

2x^2 + 3x - 2

This factorises to

(2x - 1)(x + 2)

This means that there will be min / max points at

x = 1/2 and x = -2

You then need to differentiate again to find the second differential which will indicate if it is a min point or a max:

24x + 18

When x = 1/2 this is 30 and therefore this is a minimum point
When x = -2 this is -30 and therefore this is a max point

To find the point of inflexion, set d^2y / dx^2 to 0

24x + 18 = 0
24x = -18
x = -3 / 4

And then you have to find the y values - i'm not doing that bit for u!!

y = 4x^3 + 9x^2 - 12x - 18

So, dy/dx = 12x^2 + 18x -12

To find the extrema set it equal to zero and solve for x.

Doing this, we see that we can divide through by the common factor of 6 which gives:

2x^2 + 3x -2 = 0

From the quadratic formula:

x =
-3 +/- SQRT[9 - (4)(2)(-2)]
--------------------------------
4

=
-3 +/- SQRT[9 + 16]
-------------------------
4

=
-3 +/- 5
----------
4

=
1/2 and -2

Plug these back into the original equation:

y = 4x^3 + 9x^2 - 12x - 18

x = 1/2 ---> 4(1/8) + 9(1/4) - 6 - 18 = 22/8 -24 = 2 3/4 - 24 = - 21 1/4
x = -2 ---> -32 + 36 + 24 -18 = 10

Since this graph clearly approaches negative infinity to the left and positive infinity to the right, these are either relative maxima or points of inflection, but not absolute extrema. I can't remember any clever technique for determining that other than sketching. Maybe someone can remember.

Double post.