Critical point

[trap]

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?

[quasar987]

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)

[trap]

thank you for the reply, I think what you said makes sense and may likely be the answer. :smile: