# For what values y is y(t) increasing/decreasing?

• Duderonimous
The conversation was looking promising until it was slightly deviated from the subject matter. In summary, the conversation discusses how to determine the values of y for which y(t) is increasing or decreasing in the equation y'=y^{3}-y^{2}-12y. The group agrees that finding the second derivative is unnecessary and that factoring the equation can provide the necessary information. The conversation also briefly touches on finding the corresponding values of t for the determined values of y.
Duderonimous

## Homework Statement

y'=y$^{3}$-y$^{2}$-12y

For what values of y is y(t) increasing and for what values is it decreasing?

## The Attempt at a Solution

I think you take the second derivative and equal it to zero to figure out the inflection pints right and then I am not so sure from there.

y''=3y$^{2}$-2y-12

0=3y$^{2}$-2y-12

y=$\frac{2\pm2\sqrt{37}}{6}$

Am I correct in this approach? I don't think I am.

Duderonimous said:

## Homework Statement

y'=y$^{3}$-y$^{2}$-12y

For what values of y is y(t) increasing and for what values is it decreasing?
For y to be either increasing or decreasing, you just need to look at whether y' is positive or negative, so the original equation gets you half-way there already - no need to differentiate again.

Try to factorize the RHS and it should become clear.

You also need to take y''. Solving for y' = 0 just gets you the values of y for which the slope is zero. To also learn whether those points are (relative) minima or maxima requires taking y'' and substituting the values of y for which y' = 0.

Yes, factor by all means.

rude man said:
You also need to take y''. Solving for y' = 0 just gets you the values of y for which the slope is zero. To also learn whether those points are (relative) minima or maxima requires taking y'' and substituting the values of y for which y' = 0.

Yes, factor by all means.
Rubbish.

oay said:
Rubbish.

I say! I think you're right!

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Odd that oay should be ruder than rude man and then rude man agrees with him! But oay is right- the second derivative is irrelevant. You are NOT asked to find "inflection points".

y'= y3- y2- 12y= y(y2- y- 12)= y(y- 4)(y+ 3), then if y< -3, all three factors are negative so y' is negative. If -3< y< 0 then y+3 is positive while y and y= 4 are both negative so y' is positive, etc.

But you would still need to find what values of t give those values of y.

Last edited by a moderator:
I agree with hallsofivy. When you're wrong... admit it! And I do.

HallsofIvy said:
But you would still need to find what values of t give those values of y.
Finding values of t wasn't in the question, though.
rude man said:
I agree with hallsofivy. When you're wrong... admit it! And I do.
And I apologize for my sharp reply of "Rubbish". I think I'd just had a bottle of Chardonnay.

oay said:
Finding values of t wasn't in the question, though.

And I apologize for my sharp reply of "Rubbish". I think I'd just had a bottle of Chardonnay.

No apology required! It was just that.

rude man said:
No apology required! It was just that.

Apt comment gracefully accepted and, shall I say, "wittily" replied to. I worry a lot less about rudeness when replying to somebody with 1600+ posts under their belt. I assume that's made them at least a little bit thick skinned. Nice exchange. I don't think there was any real rudeness intended and better yet, none recieved.

## 1. What is the definition of increasing/decreasing in terms of a function?

Increasing refers to the trend of the function going up as the input variable increases, while decreasing refers to the trend of the function going down as the input variable increases.

## 2. How can we determine if a function is increasing or decreasing without graphing?

If the function is in the form of a polynomial or a rational function, we can find the derivative and set it equal to 0. If the derivative is positive, then the function is increasing, and if it is negative, then the function is decreasing.

## 3. Can a function be both increasing and decreasing at the same time?

No, a function cannot be both increasing and decreasing at the same time. It can only be either increasing or decreasing.

## 4. What is the role of the second derivative in determining the increasing/decreasing behavior of a function?

The second derivative can tell us about the concavity of the function. If the second derivative is positive, the function is concave up, and if it is negative, the function is concave down. This information, along with the sign of the first derivative, can help us determine if the function is increasing or decreasing.

## 5. Can a function be increasing or decreasing at a specific point?

No, a function cannot be increasing or decreasing at a specific point since these terms describe the trend of the whole function over a certain interval. However, we can talk about the instantaneous rate of change, or the slope of the tangent line, at a specific point on the function.

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