Second derivative test I

1
Second derivative test (I)

• 1. Take the derivative of the function, f(x)
• 2. Set f(x) 0 and solve for x.
• Example If f(x) 4/3x32x2-3x1
• f (x) 4x2 4x 3 0
• (2x - 1)(2x 3) 0
• so x ½ or x-1½

2
Second derivative test (II) f (x) 4x2 4x
3

• 3. f(x) 8x 4
• 4. f(-3/2) -8 lt0 f(1/2) 8 gt 0
• 5. max _at_ -3/2 min _at_ 1/2

3
Concavity f (x) 8x 4

• f(x) 8x 4 0 when x -½
• f(-1) -4 lt 0 concave down on (-oo, -½ )
• f(0) 4 gt 0 concave up on ( -½ , oo)
• X -½ is an inflection point

4
Second derivative test (I)

• 1. Take the derivative of the function, f(x)
• If f(x) x/(1 x2)
• f (x)
• 2. Solve for x
• x 1 or x -1

5
Second derivative test (II) f (x)
(1-x2)/(1x2)2

• 3.
• 4. f(-1) 8/16 f(1) -8/16
• 5. relative min _at_ -1 max _at_ 1

6
Second derivative test (I)

• 1. Take the derivative of the function, f(x)
• 2. Set f(x) 0
• If f(x) x 9/x2 x 9x-1 2
• f(x) 1 9x-2 1 9/x2
• 0
• Numerator 0 when x 3 or x -3

7
Second derivative test (II) f (x)

• 3.
• 4. f(-3) -54/81 f(3) 54/81
  
• 5. max _at_ x -3 min _at_ x 3

8
Second derivative test (I)

• 1. Take the derivative of the function, f(x)
• 2. Set f(x) 0
• If f(x) (x 2)4
• f(x) 4(x 2)3(1)
• x 2 0
• when x 2

9
Second derivative test (II) f (x) 4(x 2)3

• 3. f (x) 12(x 2)2
• 4. f(2) 0
• 5. Test fails

10
First derivative test (II) f (x) 4(x 2)3

• f(0) -32 f(3)
  4(3-2)3 gt 0
• decreasing increasing
• relative min _at_ 2