# Prove Monotony of Function: $f$ Strictly Decreasing

• MHB
• awsomeman
In summary, a function is strictly decreasing if its output values decrease as its input values increase, and this can be proven by showing that for any two input values, x1 and x2, where x1 < x2, the corresponding output values, f(x1) and f(x2), satisfy the inequality f(x1) > f(x2). A function cannot be both increasing and decreasing, but it can be neither increasing nor decreasing. The main difference between strictly decreasing and decreasing functions is the strict requirement for output values to decrease in a strictly decreasing function. Proving the monotony of a function is important because it provides useful information about the function's behavior and is necessary for determining its continuity and differentiability.
awsomeman
Let $f$ be differentiable from $(-\inf,0)$ to $(0,\inf)$ and let $f'(x)<0$ for all real numbers except 0 and $f'(0)=0$. Prove that f is strictly decreasing.

You might want to begin by stating the definition of a decreasing function. Then consider some examples.

I would use "proof by contradiction". Suppose f is NOT strictly decreasing. Then there exist a, b, b> a, such that $$f(b)\ge f(a)$$. So $f(b)- f(a)\ge 0$. Since b> a, b- a> 0 so $\frac{f(b)- f(a)}{b- a}\ge 0$. Now use the "mean value" property.

## What does it mean for a function to be strictly decreasing?

A function is strictly decreasing if its output values decrease as the input values increase. In other words, as the input increases, the output decreases.

## How can I prove that a function is strictly decreasing?

To prove that a function is strictly decreasing, you must show that for any two input values, the output value of the first input is greater than the output value of the second input. This can be done through various methods, such as using the derivative or using the definition of a strictly decreasing function.

## What is the difference between a strictly decreasing function and a non-decreasing function?

A strictly decreasing function is one in which the output values must decrease as the input values increase. In contrast, a non-decreasing function is one in which the output values can either stay the same or increase as the input values increase.

## Can a function be strictly decreasing on some intervals and not others?

Yes, a function can be strictly decreasing on some intervals and not others. This means that the function may have regions where the output values decrease as the input values increase, but it may also have other regions where the output values do not follow this pattern.

## What are some real-life examples of strictly decreasing functions?

Some real-life examples of strictly decreasing functions include the temperature of an object as it cools down, the amount of money in a bank account as it is withdrawn, and the population of a species as it becomes endangered.

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