4.4: The Shape of a Graph

The shape of the graph is either up or down,
the shape of a smile or the shape of a frown.
The graph will contain a few points of inflection,
it's really easy to do if you follow directions.
To find the shape use the second derivative test,
Using a sign chart will end your quest.


Problem:
1. Using the equation f(x)=x^4-4x^3, find the second derivative, the critical points at the second derivative and where the shape of the graph is concave up/down.

f(x)=x^4-4x^3

f'(x)=4x^3-12x^2

f''(x)=12x^2-24x

f''(x)=12x^2-24x=0
12x(x-2)=0
12x=0; x=0
x-2=0; x=2

f(0)=0^4-4(0)^3=0. The first critical point is (0,0)
f(2)= 2^4-4(2)^3=-16. The second critical point is (2,-16)

To determine the sign of the graph before and after each critical point, I picked the numbers -1, 1, and 3. At f''(-1) the slope of the line is positive, at f''(1), the slope of the line is negative and at f''(3) the slope of the line is positive. This indicates that the shape of the graph at -1 is concave down and at 1 it is concave up.