# Calculus Problem - Properties of a Function

• Joschua_S
In summary, the problem is to show that the function f(x) = 0 is true for all x in the interval [0,1], given that f is a continuous and differentiable function on (0,1) with the properties f(0) = 0 and |f'(x)| ≤ M|f(x)| for all x in (0,1). The hint provided is to consider the set D of points in [0,1] where f(t) = 0 for all t in [0,x] and show that the supremum of this set is 1. The solution involves using the mean value theorem and considering smaller intervals to show that f(x) = 0 for all x in [0
Joschua_S
Hello

I have a mathematical problem that I could not solve. Could you please give me some hints how to solve it?

Let $f: [0,1] \rightarrow \mathbb{R}$ be a continuous and on $(0,1)$ a differentiable function with following properties:

a) $f(0) = 0$
b) there exists a $M>0$ with $|f'(x)| \leq M |f(x)|$ for all $x \in (0,1)$

Now the problem is: Show that $f(x) = 0$ is true for all $x \in [0,1]$

There is a hint given but it doesn't help me The hint is: Consider the set $D = \{ x \in [0,1]: ~ f(t) =0$ for $t \in [0,x] \}$ and show that the the supremum of this set is $1$.

Thanks for help
Greetings

What if we look at the interval ##I = [0, \frac{1}{2M}]## or ##I = [0, 1]##, whichever is smaller. This ensures that any point in ##I## will be no larger than ##\frac{1}{2M}##.

As ##f## is continuous on ##I##, so is ##|f|##, so ##|f|## achieves a maximum at some point ##x \in I##. If ##x = 0## then we're done. Otherwise, we can apply the mean value theorem to ##f## on ##[0,x]## and, with a bit of work, derive a contradiction.

P.S. The above will show that under the given hypotheses, we will have ##f(x) = 0## for all ##x \in I##. If ##I = [0,1]## then we're done. Otherwise, this shows that ##f(x) = 0## for all ##0 \leq x \leq \frac{1}{2M}##. In that case, you can repeat the argument for ##[\frac{1}{2M}, \frac{2}{2M}]## and so forth until you have covered all of ##[0,1]##.

## What is a calculus problem on the properties of a function?

A calculus problem on the properties of a function involves using calculus concepts and techniques to analyze and understand the behavior of a mathematical function. This can include finding the function's domain and range, determining its critical points and extrema, and evaluating its derivative and integral.

## What are some common properties of a function that are studied in calculus?

Some common properties of a function that are studied in calculus include continuity, differentiability, concavity, and the behavior of the function at its endpoints. These properties can provide important information about the function and how it changes over its domain.

## How is the derivative of a function related to its properties?

The derivative of a function is a fundamental tool in studying its properties. It can be used to determine the slope of the function at any point, identify critical points and extrema, and determine the function's concavity. The derivative also plays a crucial role in optimization, as it can be used to find the maximum or minimum value of a function.

## What is the significance of finding the domain and range of a function?

The domain and range of a function are important properties that can help us understand how the function behaves and what values it can take on. The domain is the set of all possible inputs for the function, while the range is the set of all possible outputs. Knowing the domain and range can also help us identify any restrictions or limitations on the function.

## How can calculus be used to solve problems involving functions?

Calculus provides powerful tools for analyzing and manipulating functions. By using concepts such as derivatives, integrals, and limits, we can solve a wide range of problems involving functions. These can include finding the maximum or minimum value of a function, determining the rate of change of a function, and calculating the area under a curve.

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