Finding Tangent Lines and Maxs/Mins

[adunbar]

Given the following table,

(and assuming f(x) and g(x) are continuous and differentiable on the closed interval [-2, 2]),

I need to find
1. if f(x) or g(x) have any horizontal tangent lines, and
2. if f(x) or g(x) have any local maximums or minimums.

I've never completed a problem like this without having a formula. Do I need to necessarily extract some sort of formula from the table in order to solve the problem? From that point this would be easy and straightforward for me. Thanks!

 

[Steven G]

Given the following table,

(and assuming f(x) and g(x) are continuous and differentiable on the closed interval [-2, 2]),

I need to find
1. if f(x) or g(x) have any horizontal tangent lines, and
2. if f(x) or g(x) have any local maximums or minimums.

I've never completed a problem like this without having a formula. Do I need to necessarily extract some sort of formula from the table in order to solve the problem? From that point this would be easy and straightforward for me. Thanks!

When does a function which is differentiable have a local max or local min?? You need to know the answer to that to do 2. And your answer can't just be when the derivative is zero. You need to think more deeply than that. Maybe looking at the data can shed some light on how you might think about this.

For 1--what is a necessary condition for a continuous differentiable function to have a horizontal tangent line??

Show us some attempt you made to solve this.

 

[adunbar]

When does a function which is differentiable have a local max or local min?? You need to know the answer to that to do 2. And your answer can't just be when the derivative is zero. You need to think more deeply than that. Maybe looking at the data can shed some light on how you might think about this.

For 1--what is a necessary condition for a continuous differentiable function to have a horizontal tangent line??

Show us some attempt you made to solve this.

Thanks for your suggestions.

So far I've attemped to use the Intermediate Value Theorem to prove that horizontal tangent lines do exist since their slopes go from positive to negative or vice-versa. I suppose that is the necessary condition for a continuous differentiable function to have a horizontal tangent line. However, while I know this can prove a horizontal tangent line exists I'm not sure the theorem will be enough to tell me exactly where they are.

Local maxs and mins occur during the highest or lowest values of the function within a specific range, so in this case between [-2,2].

 

[Ishuda]

Given the following table,

(and assuming f(x) and g(x) are continuous and differentiable on the closed interval [-2, 2]),

I need to find
1. if f(x) or g(x) have any horizontal tangent lines, and
2. if f(x) or g(x) have any local maximums or minimums.

I've never completed a problem like this without having a formula. Do I need to necessarily extract some sort of formula from the table in order to solve the problem? From that point this would be easy and straightforward for me. Thanks!

What I would do is draw a (very) rough sketch of the graph. Start by just plotting the points (the value of the function). next, add a short line through the point indicating the direction the curve is headed (the sign of the derivative). Now, roughly fill in the rest of the curve. Since the function is continuous and differentiable on -2,2 what does that tell you

EDIT: The intermediate value theorem could, as you indicated, be used to formalize your answer.

 

[adunbar]

What I would do is draw a (very) rough sketch of the graph. Start by just plotting the points (the value of the function). next, add a short line through the point indicating the direction the curve is headed (the sign of the derivative). Now, roughly fill in the rest of the curve. Since the function is continuous and differentiable on -2,2 what does that tell you

EDIT: The intermediate value theorem could, as you indicated, be used to formalize your answer.

Alright, I drew a couple rough sketches: one of the graph of f(x), and the other a graph of f'(x). After looking at the images I want to conclude (for the graph f(x)) that the local max is 5 when x=0 and the local min is -9 when x=-1 (which could be concluded simply from looking at the graph, though it is more convincing with the visual). However, how can I be certain that when x=-1.5 (for example) that there isn't a lower local minimum? Perhaps I am overthinking this.

 

[adunbar]

After looking into the problem further I believe I have found an applicable solution. I'm gonna meet with my professor tomorrow to confirm. Thanks!