Graph continuous function

[john3j]

How do I graph a continuous function that meets all of these criteria?

f(10)=5
f'(5)=0
f''(5)=10
f''(x)<0 if x>10
f''(x)>0 if x<10

If someone could explain this for a complete idiot, thats what I am, so it would be helpful.

Thanks,
John

 

[JeffM]

How do I graph a continuous function that meets all of these criteria?

f(10)=5
f'(5)=0
f''(5)=10
f''(x)<0 if x>10
f''(x)>0 if x<10

If someone could explain this for a complete idiot, thats what I am, so it would be helpful.

Thanks,
John

What is perhaps strange to you about this problem is that an infinite number of functions meet these criteria. The problem is asking you to define one of those functions and draw its graph. What this means in practice is for you to choose the missing criteria (actually you only need to pick one criterion in this case) and sketch the curve of the resulting function.

What do you know about the graph of the function given that f(10) = 5?

If a function has a first derivative equal to 0 at a point and a positive second derivative at the same point, what does that tell you about the function at that point?

Is a function everywhere doubly differentiable also continuous everywhere?

If a function is everywhere doubly differentiable and the second derivative at a point is negative to the right of that point but positive to the left of that point, what is the value of the second derivative at that point? What does that mean?

 

[HallsofIvy]

First there exist an infinite number of continuous functions that meet these criteria. That's why you are asked to find "a" continuous function, not "the" continuous function.

How do I graph a continuous function that meets all of these criteria?

f(10)=5
f'(5)=0
f''(5)=10
f''(x)<0 if x>10
f''(x)>0 if x<10

Do you understand what the derivative are and what they tell us about a graph? If you do, just apply those ideas to this. For example, f'(5)= 0 means that the tangent to the curve at x= 5 is horizontal. f''(x)< 0 if x> 10 means the graph is convex downward and f''(x)> 0 if x> 10 means it is concave upward.

It is, of course, possible to find a specific function
These last two imply that f''(x)= 0 at x= 10 so we are really given 4 specific values. That, in turns, means that we can fit a cubic function to them. Assume the function is \(\displaystyle f(x)= ax^3+ bx^2+ cx+ d\). Then f(10)= 1000a+ 100b+ 10c+ d= 5, \(\displaystyle f'(x)= 3ax^2+ 2bx+ c\) so f'(5)= 75a+ 10b+ c= 0. f''(x)= 6ax+ 2b so f''(5)= 30a+ 2b= 0. And \(\displaystyle f''(10)= 300a+ 20b+ c= 0\). Solve those four equations for a, b, c, and d.

If someone could explain this for a complete idiot, thats what I am, so it would be helpful.

Thanks,
John

 

[john3j]

First there exist an infinite number of continuous functions that meet these criteria. That's why you are asked to find "a" continuous function, not "the" continuous function.

These last two imply that f''(x)= 0 at x= 10 so we are really given 4 specific values. That, in turns, means that we can fit a cubic function to them. Assume the function is \(\displaystyle f(x)= ax^3+ bx^2+ cx+ d\). Then f(10)= 1000a+ 100b+ 10c+ d= 5, \(\displaystyle f'(x)= 3ax^2+ 2bx+ c\) so f'(5)= 75a+ 10b+ c= 0. f''(x)= 6ax+ 2b[/tex] so f''(5)= 30a+ 2b= 0. And \(\displaystyle f''(10)= 300a+ 20b+ c= 0\). Solve those four equations for a, b, c, and d.

If someone could explain this for a complete idiot, thats what I am, so it would be helpful.

Thanks,
John

[/QUOTE]

HallsofIvy,

I appreciate your responses, but I am confused. I just have to sketch the continuous function with all of those criteria met. With that being said, I dont understand the part about writing cubic functions for this problem. I am sorry for being difficult, but is there any way you could dumb this down a little more for me? In the case of f(10)=5, does that mean that 10 is point on the y axis and 5 is the point on the x axis? Is there a way to sketch this or another example within this forum?

I appreciate the help.

Thanks,
John

 

[JeffM]

I am confused. I just have to sketch the continuous function with all of those criteria met. With that being said, I dont understand the part about writing cubic functions for this problem. I am sorry for being difficult, but is there any way you could dumb this down a little more for me? In the case of f(10)=5, does that mean that 10 is point on the y axis and 5 is the point on the x axis? Is there a way to sketch this or another example within this forum?

I appreciate the help.

Thanks,
John

The conventional notation is y = f(x). If you do not understand function notation, you are far from ready for calculus.

f(10) = 5 means that when you sketch your graph, it must include the point (10, 5). This part of the exercise is basic algebra.

Both Halls and I tried to explain to you that many (in fact, an infinite number) of functions will meet the criteria specified. What Halls did was to show you how to find the simplest polynomial that meets all the criteria. It happens to be a specific cubic. That makes the problem concrete: simply sketch the graph of that cubic.

The alternative is to use your knowledge of calculus and just think about what the graph of any differentiable function looks like generically around a point where the first or second derivative has certain properties. For example, what is happening to a curve close to a point where the first derivative is zero and the second derivative is positive. Generically, what does that look like?