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trap
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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?
Local Max & Min of Continuous Function f on [a,b]
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In summary, if a function f is continuous on [a,b] and for some c in (a,b), f(c) is both a local maximum and a local minimum, it can be inferred that the function is constant on [a,b] since for all real numbers we can only have one of the three <, > or =, it will be that f(x) = f(c) near c. However, this is only a student's perspective and should not be taken too seriously.
- #2
quasar987
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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)
(but I'm just a student like you so don't take this too seriously)
- #3
trap
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thank you for the reply, I think what you said makes sense and may likely be the answer. 
FAQ: Local Max & Min of Continuous Function f on [a,b]
What is a local max/min of a continuous function?
A local max/min of a continuous function refers to the highest or lowest point on a graph within a specific interval. It is also known as a relative max/min because it is only compared to nearby points on the graph.
How do you find the local max/min of a continuous function?
To find the local max/min of a continuous function, you must first take the derivative of the function and set it equal to 0. Then, solve for the variable to find the critical points. Next, plug the critical points into the original function to determine the corresponding y-values. The highest y-value is the local max and the lowest y-value is the local min.
What is the difference between local and global max/min?
A local max/min is the highest/lowest point on a graph within a specific interval, while a global max/min is the highest/lowest point on the entire graph. This means that a global max/min can also be a local max/min, but not all local max/min are global max/min.
Can a continuous function have multiple local max/min?
Yes, a continuous function can have multiple local max/min. This can occur when the graph has multiple peaks and valleys within a specific interval.
How do local max/min relate to the overall shape of a continuous function?
The local max/min are important points that can reveal the overall shape of a continuous function. They can indicate the direction of the function, whether it is increasing or decreasing, and the presence of any peaks or valleys. By analyzing the local max/min, you can get a better understanding of the behavior of the function as a whole.