Increasing and Decreasing Functions

[courtrigrad]

Hello all

$$y = \frac {1}{4} x^4 - \frac {2}{3}x^3 + \frac {1}{2}x^2 - 3$$

Find the exact intervals in which the function is

(a) increasing
(b) decreasing
(c) concave up
(d) concave down

Then find

(e) local extrema
(f) inflection points

So I found $$\frac {dy}{dx} = x^3 - 2x^2 + x$$.

I set $$\frac {dy}{dx} = x^3 - 2x^2 + x = 0$$ yielding $$x = 0 , 1$$ with 1 as a repeated root.

However after I find my critical points, how would I determine if the function is increasing or decreasing? Would I just choose a point less than, contained, and greater than the critical points and see if the sign changes from positive to negative, vice versa or not at all? If its - + - for example. the function is increasing than decreasing. But how do you know which intervals this happens in?

Okay, to determine concavity I find $$\frac {d^2y}{dx^2} = 3x^2 - 4x + 1$$. I set this equal to 0, find the value of x. Now how would I use these values to find if the function is concave up or down? I know that these points could be inflection points, but how do you know? I know if second derivative is negative, function is concave down and the same is true for the opposite. But what if the function changes concavity? How would you find the exact intervals where it is concave up or concave down?

Finally how do you find local extreme values? Do you just use the critical points? What is the difference in finding the relative extrema versus the local extrema?

Thanks a lot for any help!

PS: If you are given a graph, how would you determine relative and local extrema without being given the function?

 

[MathStudent]

Ok you kind of have the right ideas:
here are some hints to help you get on track:

- You have the first derivative, how can this be interpreted graphically? How does this relate to increasing/ decreasing functions?

- What does the second derivative represent for a function of x? What does the second derivative tell you about the first derivative?

- What can you say about the above properties at a point that is a local max/min of a function?

- As far as inflection points go,,, first try to get A-E then worry about this.

I realize these hints don't offer a lot help at first, but they are an attempt to get you to explain what you know and help you get an idea of how to approach the problem. As this is a very lengthy question, work on only one part at a time( say part a ), share your thoughts and any work you've done, and we'll try to help you along the way.

PS: If you are given a graph, how would you determine relative and local extrema without being given the function?

You can't. Graphs can give us a visual interpretation of how a function behaves over its domain, and you may be tempted to determine the extrema by eye, however this can only give you an approximation. ( I assume here by relative extrema you mean absolute extrema, as i understand relative and local extrema to be the same)

Keep at it

 

[wais]

To determine if the function is increasing or decreasing, you can use the first derivative test. This means that you can pick a point in each interval and plug it into the first derivative. If the result is positive, then the function is increasing in that interval. If it is negative, then the function is decreasing in that interval.

To determine concavity, you can use the second derivative test. This means that you can pick a point in each interval and plug it into the second derivative. If the result is positive, then the function is concave up in that interval. If it is negative, then the function is concave down in that interval.

To find local extrema, you can use the critical points (where the first derivative is equal to 0 or undefined) and plug them into the original function. The resulting points will be the local extrema.

The difference between local and relative extrema is that local extrema are the highest or lowest points in a small interval, while relative extrema are the highest or lowest points in the entire domain of the function.

If you are given a graph, you can determine the relative and local extrema by looking at the highest and lowest points on the graph. These will be the relative extrema. To find the local extrema, you can zoom in on the graph and identify the highest and lowest points in a small interval.

 

[wais]

To determine if a function is increasing or decreasing, you can use the first derivative test. This means evaluating the first derivative at certain points and determining the sign. If the derivative is positive, the function is increasing and if it is negative, the function is decreasing. You can also use the second derivative test to determine concavity. If the second derivative is positive, the function is concave up and if it is negative, the function is concave down.

To find local extrema, you would use the critical points that you found by setting the first derivative equal to zero. The difference between local and relative extrema is that local extrema are the highest or lowest points in a specific interval, while relative extrema are the highest or lowest points in the entire domain of the function. To determine inflection points, you would set the second derivative equal to zero and solve for x. These points indicate where the concavity of the function changes.

If you are given a graph, you can determine the relative and local extrema by looking at the highest and lowest points on the graph. These points will correspond to the relative and local extrema of the function. However, without the function, it would be difficult to determine the exact values of these extrema.