Local Extrema & Mean Value Theorem

1
Local Extrema Mean Value Theorem

• Local Extrema
• Rolles theorem What goes up must come down
• Mean value theorem Average velocity must be
  attained
• Some applications Inequalities, Roots of
  Polynomials

2
Local Extrema
3
Example

• Let f(x) (x-1)2(x2), -2 ? x ? 3
• Use the graph of f(x) to find all local extrema
• Find the global extrema

4
Example

• Consider f(x) x2-4 for 2.5 ? x lt 3
• Find all local and global extrema

5
Fermats Theorem

• Theorem If f has a local extremum at an
  interior point c and f? (c) exists, then f? (c)
  0.
• Proof Case 1 Local maximum at interior point c
• Then derivative must go from ?0 to ?0 around c
• Proof Case 2 Local minimum at interior point c
• Then derivative must go from ?0 to ?0 around c

6
Cautionary notes

• f? (c) 0 need not imply local extrema
• Function need not be differentiable at a local
  extremum (e.g., earlier example x2-4)
• Local extrema may occur at endpoints

7
Summary Guidelines for finding local extrema

• Dont assume f? (c) 0 gives you a local extrema
  (such points are just candidates)
• Check points where derivative not defined
• Check endpoints of the domain
• These are the three candidates for local extrema
• Critical points points where f? (c)0 or where
  derivative not defined

8
What goes up must come down

• Rolles Theorem Suppose that f is
  differentiable on (a,b) and continuous on a,b.
  If f(a) f(b) 0 then there must be a point c
  in (a,b) where f ?(c) 0.

9
Proof of Rolles theorem

• If f 0 everywhere its easy
• Assume that f gt 0 somewhere (case flt 0 somewhere
  similar)
• Know that f must attain a maximum value at some
  point which must be a critical point as it cant
  be an endpoint (because of assumption that f gt 0
  somewhere).
• The derivative vanishes at this critical point
  where maximum is attained.

10
Need all hypotheses

• Suppose f(x) exp(-x).
• f(-2) f(2)
• Show that there is no number c in (-2,2) so that
  f ?(c)0
• Why doesnt this contradict Rolles theorem

11
Examples

• f(x) sin x on 0, 2?
• exp(-x2) on -1,1
• Any even continuous function on -a,a that is
  differentiable on (-a,a)

12
Average Velocity Must be Attained

• Mean Value Theorem Let f be differentiable on
  (a,b) and continuous on a,b. Then there must
  be a point c in (a,b) where

13
Idea in Mean Value Proof

• Says must be a point where slope of tangent line
  equals slope of secant lines joining endpoints of
  graph.
• A point on graph furthest from secant line works

14
Consequences

• f is increasing on a,b if f ?(x) gt 0 for all x
  in (a,b)
• f is decreasing if f ?(x) lt 0 for all x in (a,b)
• f is constant if f ?(x)0 for all x in (a,b)
• If f ?(x) g ?(x) in (a,b) then f and g differ
  by a constant k in a,b

15
Example
16
More Consequences of MVT
17
Examples

• Show that f(x) x32 satisfies the hypotheses of
  the Mean Value Theorem in 0,2 and find all
  values c in this interval whose existence is
  guaranteed by the theorem.
• Suppose that f(x) x2 x 2, x in -1,2. Use
  the mean value theorem to show that there exists
  a point c in -1,2 with a horizontal tangent.
  Find c.

18
Existence/non-existence of roots

• x3 4x - 1 0 must have fewer than two
  solutions
• The equation 6x5 - 4x 1 0 has at least one
  solution in the interval (0,1)