Calculus I 231

1
Calculus I- 231

• Increasing/Decreasting Test, First Derivative
  Test,
• Concavity Test and the Second Derivative Test
• Section 4.3

2
Increasing Decreasing Test

• If f(x) gt 0 on an interval I, then f is
  increasing on that interval.
• If f(x) lt 0 on an interval I, then f is
  decreasing on that interval.
• Note Algebraically to determine when a
  function is decreasing, just build a sign chart
  for f

3
Example

• Find where the function
• f(x) x3 1.5x2
• is increasing and where it is decreasing.

4
First Derivative Test

• Suppose that c is a critical number of a
  continuous function f.
• Case 1 If f changes signs at c from negative
  to positive, then f has a relative minimum at c.
• Case 2 If f changes signs at c from positive
  to negative, then f has a relative maximum at c.
• Case 3 If f does not change signs at c, then
  f does not have either a relative maximum or
  relative minimum at c.

5
Example

• Find the local minimum and local maximum
  values of the function
• f(x) (x2)2 (x-1)

6
Example

• Find the local minimum and local maximum values
  of the function
• f(x) (x4 1)/x2

7
What is concavity? What is an inflection point?

• A function is said to be concave up on an
  interval I, if f is increasing on I, and it is
  said to be concave down on an interval I, if f
  is decreasing on I.
• The point where a function changes concavity
  is where the graph of the function has an
  inflection point.

8
Concavity Test

• If f(x) gt 0 on an interval I, then f is
  increasing, and thus f is concave up on I.
• If f(x) lt 0 on an interval I, then f is
  decreasing, and thus f is concave down on I.

9
Example

• Determine where the function
• f(x) x4 4x3
• is concave up and where it is concave down and
  find all of its inflection points.

10
Second Derivative Test

• Suppose that f is continuos at c, where c is
  a critical number for f. Then
• Case 1 If f(c) gt 0, then f has a local
  minimum at c.
• Case 2 If f(c) lt 0, then f has a local
  maximum at c.
• Case 3 If f(c) 0, then we must go back to
  the FDT to decide what if anything occurs at c.

11
Example

• Use the SDT to find the relative extreme
  values of f(x) -3x55x3