Concavity and the Second Derivative Test

Concavity and the 2nd Derivative Test

New Vocabulary:

• Inflection Point: 
• Point where the concavity of the graph changes sign.
• Found where f''=0 or undefined
• Concavity
• The curve or shape of a line
• f''>0 the function is concave up (smily face)
• f''<0 the function is concave down (frowny face)

Example:

Determine the open interval on which f(x) is concave up and/or concave down, and all inflection points. f(x)=6 / (x^2+3)

ONE: Find first and second derivative

f(x)=6/(x^2+3)                        **Rearrange function so it is easier to take the first derivative
      =6(x^2+3)^-1

f'(x)= -6 (x^2+3)^-2 (2x)        **Take first derivative and simplify
       = -12x / (x^2+3)^2

f''(x)= [(x^2+3)^2 (-12) - (-12x) (2) (x^2+3) (2x)] / (x^2+3)^4         **Take the derivative
       = (x^2+3) (-12(x^2+3) + 48x^2) / (x^2+3)^4          **Factor out (x^2+3) then cancel with bottom
       = -12 [(x^2+3) - 4x^2] / (x^2 +3)^3                                            ** Factor out -12
       = -12 (-3x^2+3) / (x^2 +3)^3                                                       ** Simply top
       = 36 (x^2-1) / (x^2+3)^3                                          ** Factor out -3 and multiply with -12

TWO: f''(x) = 0 or undefined

f''(x)= 36 (x^2-1) / (x^2+3)^3

36 (x^2-1) = 0                         ** Set top equal to zero to find POSSIBLE inflection points
x= -1, 1                                   ** POSSIBLE Inflection points

Bottom is never undefined! (x^2 +3)^3 can never equal zero

THREE: Plug inflection points into original to find the y-values to the point (use calc.)

f(1) = 6 / (1^2 + 3)^2 = 3/2
f(-1) = 6 / (-1^2 +3)^2 = 3/2

FOUR: Create a number line to test the inflection points

1. Test x= -2, 0, 1 in f''(x) to see if positive or negative (shown below test number)
2. Where f''(x)>0 the function is concave up 
3. Where f''(x)<0 the function is concave down

FIVE: Answer the question

Concave up: (1, infinity) and (- infinity, -1)
Concave down: (-1,1)
Inflection Points: (1, 3/2) and (-1, 3/2)