Local Max & Min of Continuous Function f on [a,b]

Click For Summary
SUMMARY

If a continuous function f on the interval [a,b] has a point c in (a,b) that is both a local maximum and a local minimum, then f must be constant on [a,b]. This conclusion arises from the definitions of local maxima and minima, which dictate that for values near c, f(x) must equal f(c). Since f cannot simultaneously be greater than and less than f(c) in the vicinity of c, the only viable option is that f(x) equals f(c) for all x in [a,b].

PREREQUISITES
  • Understanding of continuous functions
  • Knowledge of local maxima and minima concepts
  • Familiarity with the properties of real numbers
  • Basic calculus principles
NEXT STEPS
  • Study the Intermediate Value Theorem in calculus
  • Explore the implications of the Extreme Value Theorem
  • Learn about the Mean Value Theorem and its applications
  • Investigate the concept of uniform continuity
USEFUL FOR

Students of calculus, mathematicians, and anyone interested in the properties of continuous functions and their extrema.

trap
Messages
51
Reaction score
0
what can be said about the function f, if f is continuous on [a,b], and for some c in (a,b), f(c) is both a local maximum and a local minimum?
 
Physics news on Phys.org
If the function isn't defined by parts on [a, b], I would say it means the function is constant on [a, b]. Because if c is a local max it means that for x near c, f(x) is smaller or equal to f(c). If additionnaly, c is a local min it means that for x near c, f(x) is greater or equal to f(c). Since for all real numbers we can only have one of the three <, > or =, it will be that f(x) = f(c) near c.

(but I'm just a student like you so don't take this too seriously)
 
thank you for the reply, I think what you said makes sense and may likely be the answer. :smile:
 

Similar threads

Graduate How to Find Critical Points of function f(x,y,z)
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad Understanding a proof of inexistence of max nor min
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Extreme value nonexistence proof
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad A correct definition of sequential right continuity of a function
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Should the function f to be continuous for applying MVTI or not?
  • · Replies 4 ·
Replies
4
Views
2K
MHB Understanding Continuous Functions: Examining f'(7) Undefined
  • · Replies 9 ·
Replies
9
Views
2K
MHB Is f(x)=ln(x)/x Increasing or Decreasing?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Continuity of Mean Value Theorem
  • · Replies 12 ·
Replies
12
Views
3K
Undergrad Doubt about theorem in Calculus on Manifolds
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad Uniform convergence of family of functions with continuous index
  • · Replies 3 ·
Replies
3
Views
3K