Make-up #4

First Derivative Test

The First Derivative Test is used for relative and absolute maximums and minimums.  The Advanced Placement test is loaded with both multiple-choice and short answer questions that involve the First Derivative Test.

Example:

Find the absolute maximums and minimums of the equation (1/3)x^3-2x^2+3x.

1. Take the derivative of the equation.

x^2-4x+3

2.  Set it equal to zero then solve for x.

x^2-4x+3=0   (x-3)(x-1)  x=3 x=1

3.  Set up intervals.

(-infinity,1) (1,3) (3, infinity)

4.  Plug in a number found within all intervals into your first derivative (relative maximums and minimums).

(-infinity, 1)=positive number      at x=1 there is a maximum
(1,3)=negative number               at x=3 there is a minimum
(3,infinity)=positive number

5.  Plug in x values to original equation to find absolute maximums and minimums.

x=1 is the absolute maximum (1,2/3)
x=3 is the absolute minimum (3,0)


Second Derivative Test

The Second Derivative Test is useful for finding concavity or absolute maximums and minimums.

Example:

Find the absolute maximum of the equation y=x^3-3x

1.  Find first derivative.

x^3-3x
3x^2-3

2.  Set equal to 0 then solve for x.

3x^2-3=0  3x^2=3  x^2=1  x=1 x=-1

3. plug into second derivative.

3x^2
6x    6(1)=6   6(-1)=-6   x=6 there is a absolute maximum.