# Increasing/Decreasing function, no max/min

• AquaGlass
In summary, the conversation discusses the rational function f ' (x) = [-(x^2) - 9]/[[(x^2) - 9]^2] and its critical numbers of 3 and -3. The speaker clarifies that these are real roots and not complex roots. They also explain that the function has no maximum or minimum points due to its vertical asymptotes at x=3 and x=-3. They further describe the behavior of the function for different ranges of x values. The conversation ends with the conclusion that there are no turning points or maximum/minimum values for this function.
AquaGlass
thanks!

Last edited:
I obtained f ' (x) = [-(x^2) - 9]/[[(x^2) - 9]^2]
I found the critical numbers of 3 and -3.
Are these real roots, or did you mean 3i, -3i?

+(x^2) - 9 has real roots 3 and -3.

-(x^2) - 9 = -((x^2) + 9) has the complex roots I stated.

I'm wondering why you find it hard to believe this has no max or min. This is a rational function that has vertical asymptotes at x= 3 and x= -3 (because the denominator factors as (x-3)(x+3)). If x< -3, all of x, x-3, and x-3 are negative so the fraction itself is negative. The graph goes from asymptotic to 0 for large negative x to going to -infinity as it approaches x= -3. If -3< x< 0, the fraction is positive. The graph goes from +infinity close to x= -3 through 0. If 0< x< 3, the fraction is negative and goes from 0 toward - infinity as x approaches 3. Finally, for x> 3 the fraction is negative and runs from near + infinity close to 3 to asymptotic to 0 as x goes to +infinity.
There are no turning points and no max or min.

## What is an increasing function?

An increasing function is a mathematical function that follows the pattern of increasing values as the input values increase. In other words, as the input values increase, the output values also increase.

## What is a decreasing function?

A decreasing function is a mathematical function that follows the pattern of decreasing values as the input values increase. In other words, as the input values increase, the output values decrease.

## What is the difference between an increasing and decreasing function?

The main difference between an increasing and decreasing function is the direction in which the output values change in relation to the input values. In an increasing function, the output values increase as the input values increase, while in a decreasing function, the output values decrease as the input values increase.

## Can a function be both increasing and decreasing?

No, a function cannot be both increasing and decreasing at the same time. A function can only follow one pattern of either increasing or decreasing values as the input values increase.

## Can a function have no maximum or minimum value?

Yes, a function can have no maximum or minimum value. This means that the function does not have a highest or lowest point, and the output values continue to increase or decrease without ever reaching a maximum or minimum value.

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