Question 1004329
Let f(x)=(2/3)x³-5x²+12x-7 
Find f'(x) and f"(x).

f'(x) = 2x²-10x+12
f"(x) = 4x-10 

Find the intervals on which f(x) is increasing or decreasing.
<pre>
Set the first derivative = 0

f'(x) = 2x²-10x+12 = 0
           x²-5x+6 = 0
        (x-3)(x-2) = 0
            x-3=0; x-2=0
              x=3    x=2

This divides the number line into three intervals

------o---o-------
      2   3

Intervals: | (-infinity,2) |   (2,3)  | (3,infinity)
Test value:|       0       |    2.5   |     4
Value of f'|      12       |    -.5   |     4
Sign of f' |       +       |     -    |     +
Conclusion |  increasing   |Decreasing| Increasing
</pre>
Find the local maximum and minimum of f(x), if any
<pre>
Since f(x) changes from increasing to decreasing at
x=2, and f(2) is defined, therefore there is a "peak",
or local maximum where x=2.
Since f(2) = 7/3, the local maximum point is (2,7/3) 

Since f(x) changes from decreasing to increasing at
x=3, and f(3) is defined, therefore there is a "valley",
or local minimum where x=3.
Since f(3) = 2, the local minimum point is (3,2)
</pre>                              
Find the intervals on which the graph of f (x) is concave up or concave down.
<pre>
Set the second derivative = 0

f"(x) = 4x-10 = 0
           4x = 10
            x = 10/4
            x = 5/2

This divides the number line into two intervals

-------o-------
      5/2

Intervals: |(-infinity,5/2)|(5/2,infinity)
Test value:|       0       |    3         |     
Value of f"|     -10       |  7.5         |     
Sign of f' |       -       |    +         |     
Conclusion |concave down   |concave up    |
</pre>
Find the inflection points of f(x), if any.
<pre>
Since f(x) changes from concave downward to concave upward
at x=5/2, and f(5/2) is defined, therefore there is an
inflection point, where x=5/2.
Since f(5/2) = 13/6, the inflection point point is (5/2,13/6).

On the graph below, the two green points are the local maximum
and minimum, and the red point between is the inflection point. 


{{{drawing(400,240,-5,5,-1,5,


green(circle(2,2.33333333,0.09),circle(2,2.33333333,0.07),circle(2,2.33333333,0.05),circle(2,2.33333333,0.03),circle(2,2.33333333,0.01)),

green(circle(3,2,11),circle(3,2,.07),circle(3,2,0.05),circle(3,2,0.03),circle(3,2,0.01)),

red(circle(2.5,2.16666667,0.09),circle(2.5,2.16666667,0.07),circle(2.5,2.16666667,0.05),circle(2.5,2.16666667,0.03),circle(2.5,2.16666667,0.01)),




graph(400,240,-5,5,-1,5,(2/3)x^3-5x^2+12x-7)   )}}}

Edwin</pre>