1. Extremas – First Derivative Test & Second Derivative Test
Calculus - Santowski
11/29/2018

2. Fast Five
1. Determine the first and second derivatives of the function f(x) = x4 – 4x3 + 4x2
2. Determine the first and second derivatives of the function f(x) = x4 – 4x3
3. Determine the first and second derivatives of the function f(x) = x2e-x
4. Determine the first and second derivatives of the function f(x) = 2sin(x) + sin2(x)
Calculus - Santowski
11/29/2018

3. Lesson Objectives
1. Explain what the First Derivative Test is and why it “works”
2. Explain what the Second Derivative Test is and why it “works”
3. Work with the FDT & SDT to classify extrema
Calculus - Santowski
11/29/2018

4. (A) First Derivative Test
Let f be a differentiable function with f '(c) = 0 then
1. If f '(x) changes from positive to negative, then f has a relative maximum at c.
2. If f '(x) changes from negative to positive, then f has a relative minimum at c.
3. If f '(x) does not change sign at c, then f has neither an maximum or minimum at c. (Stationary point)
Calculus - Santowski
11/29/2018

5. (A) First Derivative Test
Use the FDT to classify all extrema of the function f(x) = x4 – 4x3 + 4x2
Calculus - Santowski
11/29/2018

6. (B) Second Derivative Test
Let f be a twice differentiable function near x = c such that f '(c) = 0. Then
1. If f ''(x) > 0 then f(c) is a relative minimum.
2. If f ''(x) < 0 then f(c) is a relative maximum.
3. If f ''(x) = 0 then use the first derivative test to classify the extrema.
Calculus - Santowski
11/29/2018

7. (B) Second Derivative Test
Your task: Write an explanation/clarification/rationalization of the FDT. Include diagrams in your explanation
Calculus - Santowski
11/29/2018

8. (B) Second Derivative Test
Use the function f(x) = x4 – 4x3 to show algebraically HOW the SDT can be used to classify the extremas as either (i) maximums, (ii) minimums, or (iii) stationary points
Calculus - Santowski
11/29/2018

9. (C) Examples
ex 1. Find and classify all extrema using FDT of f(x) = 3x5 - 25x3 + 60x.
ex 2. Find and classify all extrema using SDT of f(x) = 3x4 - 16x3 + 18x2 + 2.
Calculus - Santowski
11/29/2018

10. (C) Examples - FDT
Ex 3. Find the intervals of increase and decrease and max/min values of f(x) = cos(x) – sin(x) on (-,)
Ex 4. Find the intervals of increase/decrease and max/min points of f(x) = x2e-x
Ex 5. Find the local and absolute maximum & minimum points for f(x) = x(ln(x))2
Calculus - Santowski
11/29/2018