What is the critical point and location of this graph?

[Femme_physics]

Homework Statement

http://img508.imageshack.us/img508/5761/calculus01question.jpg



Do they mean 1/x? I'm not sure what they mean by 1 over f(x)

It seems to me I just have to answers questions based on the graph. From some reasons my answers are off compared to the answer book.

The questions are:

A) Find the critical point type f(x)
B) The location of f(x) critical point
C) The location of f(x) graph where it goes through the x axis, and the y-axis

The Attempt at a Solution

A) Maxima
B) (3, -2)
C) (0, 0.25)
The function doesn't appear to go through the x axis.



Weird, the answer book says it's a min not and max. Apparently I'm wrong everywhere. Can anyone help me with it?
b]

 

[hunt_mat]

I don't speak arabic but I can comment about your question. Of course 1/f(x) is a function! Take for example f(x)=\sin x Then 1/f(x)=\cosec x which is a well defined function away from the zeros of sin x. Critical point of f(x) are in general critical points of 1/f(x) because:

<br /> \left(\frac{1}{f(x)}\right) &#039;=-\frac{f&#039;(x)}{(f(x))^{2}}<br />

 

[Femme_physics]

Er, what does Arabic has to do with it? And the text isn't even Arabic. And I translated the text. Or maybe not, but it just says "This is the graph of the function 1/f(x)"

I did translate the questions.

So by saying 1/f(x) they really mean 1/(some function)?

 

[hunt_mat]

Oh right, the text looked like arabic to me.

They are indeed just defining a new function g(x)=1/f(x).

 

[Femme_physics]

It's Hebrew :)

They are indeed just defining a new function g(x)=1/f(x).

Ah...so we don't really know what the function is. But by looking at the graph we can deduce a lot of info. Can you help me understand what's wrong with my answers, though?

 

[hunt_mat]

Okay, I showed that critical points of 1/f(x) are critical points of f(x), so there is a critical point at (3,-2). As we move away from this point, the function 1/f(x) becomes more negative. At the critical point f(x) has the value -1/2. If we move out from the critical point there will be a point at which 1/f(x)=-3 and so f(x)=-1/3, so we have to ask ourselves is what is bigger, -1/2 or -1/3?

For c) you have given the point where 1/f(x) hits the y axis. For hitting the x-axis, f(x)=0 at this point, what will 1/f(x) be?

 

[I like Serena]

The graph you have is of the function 1/f(x).
The graph you need is of the function f(x).

I would suggest you try to draw the graph of f(x).
This means that for a number of x values, you look up the corresponding y-value in the current graph, calculate 1/y, and plot that.

I think you'll see then what your answers should be.

 

[Femme_physics]

Maybe I have some gaps in my calculus understanding.

The graph you have is of the function 1/f(x).
The graph you need is of the function f(x).

See, I don't get why I need a different graph from the one I'm being asked about.

I would suggest you try to draw the graph of f(x).

Isn't the graph I need already drawn to me?

Okay, I showed that critical points of 1/f(x) are critical points of f(x), so there is a critical point at (3,-2). As we move away from this point, the function 1/f(x) becomes more negative. At the critical point f(x) has the value -1/2. If we move out from the critical point there will be a point at which 1/f(x)=-3 and so f(x)=-1/3, so we have to ask ourselves is what is bigger, -1/2 or -1/3?

For c) you have given the point where 1/f(x) hits the y axis. For hitting the x-axis, f(x)=0 at this point, what will 1/f(x) be?

I'm trying to follow you but I still don't see a function in all that, I just see a graph of a supposed existing function that isn't given to me. I'm very confused. Maybe my calculus gaps are too deep to understand? I thought I'm supposed to be given a function, something like X^3 -16x +33. Here I don't see a function, it just says "here is a function but we're not going to tell you and here's its graph"... this is very confusing. Sorry if I can't relate to your answers. Thanks for trying though.

 

[hunt_mat]

The function 1/f(x) as drawn seems to diverge to -infinity at the points x=2 and x=4, so that would indicate that at those points f(x) must be zero, so you have two zeros of f(x) right there.

 

[HallsofIvy]

Giving you the graph gives you the function- at least to the accuracy at which you can read the graph. There is no "Calculus" required here, only the ability to read a graph.

At a number of different "x" values, you can read the corresponding "f(x)" from the graph. Use that to find 1/f(x). Mark the point (x, 1/f(x)) for those same "x" values and then draw a smooth curve through them to sketch the graph of y= g(x)= 1/f(x).

For example, f(0)= 0.25 so g(0)= 4. The point (0, 4) is on the graph of y= g(x). f(1) looks to be about 3/4 so g(1) is about 4/3. Marking the point (1, 4/3) won't be far wrong. It looks like f goes to infinity as x goes to 2 so g must go to 0 there. Mark (2, 0) on the graph of y= g(x). Similarly, f(3)= -3 so g(3)= -1/3. f goes to infinity as x goes to 4, so g(4)= 0. f(5) is about 3/4, again, so g(5) is about 4/3, f(6)= 1/4 so g(6)= 4.

Or are you given a graph of y= 1/f(x) and asked to graph y= f(x)? If so, just swap "f" and "g" in what I said above. The idea is the same.

 

[I like Serena]

Let me see if I can make it a bit clearer.

I've created the following table and figure to explain:

The table gives a number of x-values and the corresponding y-values (blue graph), such that the corresponding graph looks a bit like the one in your problem.
The y-values are the ones that correspond to the y=1/f(x) graph which is given in your problem.

The third column contains the 1/y values of the second column (red graph), which will make them correspond to y=f(x).

Your problem asks for answers for the red graph.
Does this help?

 

[hunt_mat]

Let me see if I can make it a bit clearer.

I've created the following table and figure to explain:
View attachment 34666

The table gives a number of x-values and the corresponding y-values (blue graph), such that the corresponding graph looks a bit like the one in your problem.
The y-values are the ones that correspond to the y=1/f(x) graph which is given in your problem.

The third column contains the 1/y values of the second column (red graph), which will make them correspond to y=f(x).

Your problem asks for answers for the red graph.
Does this help?

Respect to you, you went to a lot of effort here!

 

[I like Serena]

Respect to you, you went to a lot of effort here!

Thanx!
Let's just hope it is appreciated.

 

[Femme_physics]

Serena! Wow. Above and beyond. I never had anything handed to me on a silver platter like that. I'm... not used to being pampered like that. You're going to spoil me rotten sis :D
Thank you. And thanks to the others for the contributions, I'm coming to with this calculus (or non-calculus in this case) thing. I've spent now a good chunk of time rereading the replies and churning over this.

So in the chart,
X represents the number line
f(x) represents the graph of the function (blue graph) WITHOUT the 1 over that function, just the function itself
AND the result in 1/f(x) is actually the graph I'm looking for. Because that's what they're telling me in the question. So, this basically boils down to a reading comprehension question, right?

In that case:

A) Minimum
B) (3, -0.5)
C) Visually I can see it's 2 and 4, although by Serena's chart it's undefined.

But is that correct?

 

[I like Serena]

Serena! Wow. Above and beyond. I never had anything handed to me on a silver platter like that. I'm... not used to being pampered like that. You're going to spoil me rotten sis :D
Thank you. And thanks to the others for the contributions, I'm coming to with this calculus (or non-calculus in this case) thing. I've spent now a good chunk of time rereading the replies and churning over this.

So in the chart,
X represents the number line
f(x) represents the graph of the function (blue graph) WITHOUT the 1 over that function, just the function itself
AND the result in 1/f(x) is actually the graph I'm looking for. Because that's what they're telling me in the question. So, this basically boils down to a reading comprehension question, right?

In that case:

A) Minimum
B) (3, -0.5)
C) Visually I can see it's 2 and 4, although by Serena's chart it's undefined.

But is that correct?

Your answers are correct.

But I'm afraid you're switching the graphs around the wrong way.
The graph given in your problem is the blue graph. Your problem states that this is the graph of 1/f(x).
However, the questions are about f(x) of which the graph is not given (for which I made the red graph).

Now, as you can see the value for x=2 of the red graph is 0, that is, f(2)=0.
This means the value of the blue graph for x=2, is 1/f(2)=1/0, which can not be defined. As you see the blue graph has no point for x=2, although the graph goes up to plus infinity, and comes back from minus infinity.

 

[Femme_physics]

But I'm afraid you're switching the graphs around the wrong way.
The graph given in your problem is the blue graph. Your problem states that this is the graph of 1/f(x).
However, the questions are about f(x) of which the graph is not given (for which I made the red graph).

See, again, reading comprehension *smacks forehead* Thank you for pointing that out Ser.

Now, as you can see the value for x=2 of the red graph is 0, that is, f(2)=0.
This means the value of the blue graph for x=2, is 1/f(2)=1/0, which can not be defined. As you see the blue graph has no point for x=2, although the graph goes up to plus infinity, and comes back from minus infinity.

I see that now :) Merci.

 

[hunt_mat]

Here are my lecture notes on the subject of calculus.

https://www.physicsforums.com/showthread.php?t=444599

 

[Femme_physics]

Did you forget to attach them?

 

[hunt_mat]

Click on the URL