# About a monotonically decreasing f, does it exist a theorem that says

• Temporaneo
In summary: The function is increasing on [a,x] and decreasing on [x,b] where x is your critical point, but the derivative to the left of that point must always be negative and to the right it must be positive. This means the function is increasing on [a,x] and decreasing on [x,b] if x is your critical point, and you can work out the details from there.
Temporaneo

## Homework Statement

I have y=2x*f(x).
f is strictly monotonically decreasing, non-negative, derivable and continuous in the close interval [0,c] with c>=1. it doesn't change its concavity in the interval, maybe beside at x=c/2. Note that 2x has the same properties but is monotonically increasing.
i've found a maximum point for y, at x=c/2, but not knowing f i was able only to demostrate that it was a local max. i want to say that it's the global max

2. The attempt at a solution

if there's a unique maximum for y in the interval [0,c], then my local max is automatically the global max

i think that is obviously true, but does it exist a theorem that says
the product of two nonnegative functions with fixed concavity, one monotonically increasing and the other decreasing in [a,b], is a function that has a single maximum point, being monotonically incresing before and monotonically decreasing after. if the max is in a or b then the new function is only increasing or decreasing
or something like that?

Last edited:
Counterexample: f(x)=1/x. Or 1/sqrt(x). Or1/(x2)

It's true in general that if a function is continuous and has only a single critical point, which is a local max or min, then that is a global max or min. Why? Because the derivative to the left of that point must always be positive (it can only change sign at a critical point) and to the right it must be negative. Which means the function is increasing on [a,x] and decreasing on [x,b] where x is your critical point, and you can work out the details from there.

Office_Shredder said:
Counterexample: f(x)=1/x. Or 1/sqrt(x). Or1/(x2)

Sorry but i don't understand how these are counterexample. Beside that they aren't derivable in 0, if i multiply them with 2x i obtain functions that the max is in a or b and the new function is only increasing or decreasing

Probably i failed to write that 0=<a<b<+inf

## What is a monotonically decreasing function?

A monotonically decreasing function is a type of mathematical function where the output decreases as the input increases. This means that as the independent variable increases, the dependent variable decreases, and the graph of the function forms a downward slope.

## What is a theorem?

A theorem is a statement that has been proven to be true using logical reasoning and accepted mathematical principles. In other words, a theorem is a mathematical fact that has been shown to be true.

## Does there exist a theorem about monotonically decreasing functions?

Yes, there is a theorem called the "Monotone Convergence Theorem" that states that if a sequence of numbers is monotonically decreasing and bounded below, then the sequence will converge to a limit. This theorem is commonly used in calculus and real analysis.

## Is there a specific proof for the existence of a theorem about monotonically decreasing functions?

Yes, there are various proofs for the Monotone Convergence Theorem, including one that uses the concept of supremum and infimum. Other proofs may use different mathematical techniques, but they all ultimately show that the theorem holds true for monotonically decreasing functions.

## What are some real-world applications of monotonically decreasing functions?

Monotonically decreasing functions can be used to model various real-world phenomena, such as population growth, decay of radioactive elements, and depreciation of assets. They are also commonly used in economics, finance, and natural sciences to analyze trends and make predictions based on data.

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