inflection points of a graph

Question

How many inflection points are in the graph of the curve f(x) = x4 ­– 4x2 ?

Answer 1

inflection points correspond to a first derivative of 0 and a second derivative of 0.

 

f(x)=4x-4x²

 

f'(x)=4-8x=0

 

x=0.5

 

f"(x)=-8

 

There is one critical point at x=0.5 and it is a maximum.

 

There are no points of inflection.

Answer 2

Points of inflection are regions where the graph changes concavity.  To find the points of inflection, we use the second derivative test. 

 

f"(x) = 0

 

4x³ - 8x = 0

 

 

Derive one more time,

 

12x² - 8 = 0

 

4(3x² - 2) = 0

 

 

3x² - 2 = 0

 

3x² = 2

 

 x = ±√(2/3)

 

 

x = -0.82

 

x = 0.82

 

 

These are our points of inflection.  Next, we need to use test points to see if the second derivative changes signs at certain points.  If they do not change signs, then they are not points inflection.  We will use x=-1, x=0, and x=1 and plug them into the second derivative.

 

 

f"(-1) = 4(3(-1)² - 2)

         = 4(1)

         = 4

 

f"(0) = 4(3(0)² - 2)

        = 4(-2)

        = -8

 

f"(1) = 4(3(1)² - 2)

        = 4(1)

        = 4

 

Since the signs change for each test point, we have 2 points of inflection.