Is this question missing a third time interval?

[cptstubing]

Homework Statement

Given critical points (3,-4) and (6,0); interval of increase (3, infinity); interval of decrease (-infinity, 3), find the local maxima/minima and sketch the graph.

Homework Equations

No relevant equations are given, I believe it's a simple sketch the graph.

The Attempt at a Solution

In every example so far in the study guide I'm using, if there is one critical point, there are two time intervals.
If there are two critical points, there are three time intervals.

In this question, I believe there should be a curve that decreases to (3, -4), and the a curve that increases from the previous point to (6,0), and then another curve that decreases after that. But no such time interval is given.

Is the question missing a time interval, or is there some way to do this?

Many thanks

 

[SammyS]

Homework Statement

Given critical points (3,-4) and (6,0); interval of increase (3, infinity); interval of decrease (-infinity, 3), find the local maxima/minima and sketch the graph.

Homework Equations

No relevant equations are given, I believe it's a simple sketch the graph.

The Attempt at a Solution

In every example so far in the study guide I'm using, if there is one critical point, there are two time intervals.
If there are two critical points, there are three time intervals.

In this question, I believe there should be a curve that decreases to (3, -4), and the a curve that increases from the previous point to (6,0), and then another curve that decreases after that. But no such time interval is given.

Is the question missing a time interval, or is there some way to do this?

Many thanks

What do the critical points signify? In other words, what is it that makes a critical point critical?

 

[cptstubing]

Min and max.
It's where f(x) or y starts to either increase or decrease.

 

[LCKurtz]

Homework Statement

Given critical points (3,-4) and (6,0); interval of increase (3, infinity); interval of decrease (-infinity, 3), find the local maxima/minima and sketch the graph.

Homework Equations

No relevant equations are given, I believe it's a simple sketch the graph.

The Attempt at a Solution

In every example so far in the study guide I'm using, if there is one critical point, there are two time intervals.
If there are two critical points, there are three time intervals.

In this question, I believe there should be a curve that decreases to (3, -4), and the a curve that increases from the previous point to (6,0), and then another curve that decreases after that.

But you are given that it is increasing for $t > 6$. What does that tell you about $(6,0)$?

 

[SammyS]

Min and max.
It's where f(x) or y starts to either increase or decrease.

Is that fundamentally what they are - or is that just one possible characteristic?

I ask because I suspect that they are simply the points at which the tangent line has slope of zero. -- Often there will be a min or max in that situation.

(Is this pre-calculus or is it Calculus?)

 

[cptstubing]

That's what I understand the question to say as well. It appears to be a parabola, but then the point (6,0) isn't a critical point, it's just a point along a line.

 

[SammyS]

It will help for you to use the "Reply" feature.

 

[cptstubing]

Is that fundamentally what they are - or is that just one possible characteristic?

I ask because I suspect that they are simply the points at which the tangent line has slope of zero. -- Often there will be a min or max in that situation.

(Is this pre-calculus or is it Calculus?)

Yes that is true as well, more true than what I said. Slope = zero at these critical points.

 

[cptstubing]

It will help for you to use the "Reply" feature.

I'm not following this and it is still unclear to me.
So the study guide is correct would you say? It has been incorrect already, and this is why I'm asking.
The line I need to sketch I would assume *should* look like a wave (given 2 critical points and 3 time intervals), and not at all like a parabola, which would have 1 critical point and 2 time intervals.

 

[Mark44]

You can have a critical point without the point being a local minimum or local maximum. The fact that the tangent line has a slope of zero doesn't guarantee either a maximum or minimum. For example, the graph of y = x³ has a critical point at (0, 0), but it has neither a maximum or minimum. What you do have here is an inflection point, as the concavity is changing from concave down for x < 0 to concave up for x > 0.

Also, you have referred to "time intervals," which is somewhat confusing, as I don't see that anything here relates to time. As far as I can tell, they are just intervals on which the function is increasing or decreasing.

This thread is "calculus-y" enough that I am moving it out of the precalc section.

 

[Pierce610]

if (6,0) is a critical point, however this is located in the interval of increase (3, infinity) of the function; but in a critical point can be only a local maximum o a minimum about the definition?

 

[cptstubing]

You can have a critical point without the point being a local minimum or local maximum. The fact that the tangent line has a slope of zero doesn't guarantee either a maximum or minimum. For example, the graph of y = x³ has a critical point at (0, 0), but it has neither a maximum or minimum. What you do have here is an inflection point, as the concavity is changing from concave down for x < 0 to concave up for x > 0.

Also, you have referred to "time intervals," which is somewhat confusing, as I don't see that anything here relates to time. As far as I can tell, they are just intervals on which the function is increasing or decreasing.

This thread is "calculus-y" enough that I am moving it out of the precalc section.

I follow now. My mind is biased toward earlier study guide questions.
If I had an equation for this I'd have gotten it.

The next question in the book is one I'm used to seeing (2 critical points, 3 intervals)

Many thanks!

 

[Mark44]

if (6,0) is a critical point, however this is located in the interval of increase (3, infinity) of the function; but in a critical point can be only a local maximum o a minimum about the definition?

A critical point is not necessarily associated with a max or min, which is what I said in post #10. I'm not sure if you are commenting or asking a question here.

 

[LCKurtz]

What do the critical points signify? In other words, what is it that makes a critical point critical?

Min and max.
It's where f(x) or y starts to either increase or decrease.

That's what I understand the question to say as well. It appears to be a parabola, but then the point (6,0) isn't a critical point, it's just a point along a line.

You are misunderstanding what a critical point is. Usually it is a point where $f'(x)=0$ meaning the curve has a horizontal tangent line. At such a point on a graph the curve may have:
1. A relative maximum
2. A relative minimum
3. What is the third possibility? This is what you are overlooking and need to know for your graph.

 

[SammyS]

Yes that is true as well, more true than what I said. Slope = zero at these critical points.

Good.

By the way, the slope of the tangent line can be zero at a point without it being a min or max. Consider the graph of f(x) = (x-1)³ +2. The slope is zero at the point (1,2) , but that's neither a min nor a max.

 

[cptstubing]

Good.

By the way, the slope of the tangent line can be zero at a point without it being a min or max. Consider the graph of f(x) = (x-1)³ +2. The slope is zero at the point (1,2) , but that's neither a min nor a max.

THanks, Sammy!

 

[cptstubing]

You are misunderstanding what a critical point is. Usually it is a point where $f'(x)=0$ meaning the curve has a horizontal tangent line. At such a point on a graph the curve may have:
1. A relative maximum
2. A relative minimum
3. What is the third possibility? This is what you are overlooking and need to know for your graph.

Got it!
Thanks!