# Proving Increasing Function of f(x)/x on ]0,∞[

• LikeMath
In summary, the question is asking to prove that the following function is increasing on ]0,\infty[. The first derivative is positive, so the original function is concave up. Drawing a graph may help solve the problem.

#### LikeMath

Hi! I have the following question.
Let $f$ be continuous on $[0,\infty[$, $f(0)=0$, $f^\prime$ exists on $]0,\infty[$, and $f^\prime$ is increasing on $]0,\infty[$.

the question is to prove that the following function is increasing that is $g(x)=f(x)/x$ on $]0,\infty[$.

I tried to show that the first derivative is positive but I did not succeed to use the monotonicity of $]f^\prime[$

If $f'$ is increasing, what does that say about the shape of $f$?

Can you say anything comparing $f'$ to $g$?

If the derivative is increasing that mean the original function is concave up, but how can that solve the question. I did not get it.

LikeMath said:
If the derivative is increasing that mean the original function is concave up...

Good first step.

2) Now, can you say anything about the comparison between $f'$ and $g$? Must one of them always be bigger than the other?

3) Is there a way to express $g'$ in terms of $f'$ and $g$?

I can't resist being... well, an economics nerd. This has a very clear meaning in public finance, for the case where $f(x)$ represents the number of tax dollars a person who makes income $x$ is required to pay. It says that if the marginal tax rate (i.e. how much you pay on your last dollar earned) is increasing, then taxation is progressive, i.e. higher income folks pay a higher overall tax percentage.

I would recommend drawing a picture if you're confused about how to start. Make up your favorite f(x) function that fits the requirements and draw a graph. The value of g(x) can be represented by the slope of a line that you can draw on your f(x) graph - if you can figure out what line it is the slope of you are well on your way to solving the problem

## 1. What is the definition of an increasing function?

An increasing function is a mathematical function where the value of the dependent variable increases as the value of the independent variable increases. In other words, as the input increases, the output also increases.

## 2. How can I prove that a function is increasing?

To prove that a function is increasing, you need to show that for any two input values, x1 and x2, where x1 < x2, the corresponding output values, f(x1) and f(x2), also follow the same relationship, f(x1) < f(x2). This can be done through various methods such as using calculus, graphing, or algebraic manipulation.

## 3. What does the function f(x)/x represent?

The function f(x)/x represents the ratio of the output value of a function to its input value. This function is often used to analyze the growth rate of a function as the input value increases.

## 4. Why is it important to prove that f(x)/x is increasing on the interval ]0,∞[?

Proving that f(x)/x is increasing on the interval ]0,∞[ is important because it demonstrates that the function is growing at a consistent rate as the input value increases. This information can be useful in various applications, such as in economics and physics, to analyze the behavior and trends of certain variables.

## 5. Can a function be increasing on one interval and decreasing on another?

Yes, a function can be increasing on one interval and decreasing on another. This is known as a non-monotonic function. It means that the function's behavior is not strictly increasing or decreasing throughout its entire domain, but rather varies on different intervals.