Question

Use the intermediate value theorem and Rolle's theorem to show that the graph f(x)=x^3+2x+k crosses the x-axis exactly once, regardless of the value of the constant k.

Answer 1

this is looks fun
i'm going to watch tv and work on it in the other room

Answer 2

hehehe :D okk just show me hw to do it

Answer 3

You use the intermediate value theorem to show that at least one zero exists.

You use Rolles theorem to show that at most one zero exists.

In conjunction, this shows that exactly one zero exists.

Answer 4

For the first part, you can pick sufficiently large values for the endpoints. Make sure that one endpoint is a large negative number while the other is a large positive number. With some reasoning, you can show that this will encapsulate every value of k.

Answer 5

are you familiar with the two theorems?

Answer 6

yess i am
intermediate value theorem if a function is continuous on [a,b] and k is any number between f(a) and f(b) then thers a point c between a nd b such that f(c) =k

and rolles theorem if f(a)=f(b) then then thers a point c such that f'(c)=0

Answer 7

right?

Answer 8

f'(x)=3x^2+2>0
means f is increasing and there are not horizontal tangents so it would have to cross x-axis once

Answer 9

that is correct.

Answer 10

so if a is some really large negative number and b is some really large positive number, then by the intermediate value theorem, there should be at least one zero. Why is that?

Answer 11

f(a)=a^3+2a+k
f(-a)=-a^2-2a+k

\[\lim_{a \rightarrow \infty}(a^3+2a+k)=\infty ,\lim_{a \rightarrow \infty}(-a^3-2a+k)=-\infty\]

Answer 12

f(a) is negative for some sufficiently large negative number 'a' and f(b) is positive for some sufficiently positive number 'b'.


So f(a) is negative and it jumps to f(b) being positive. This means that f(x)=0 somewhere in the middle.

Answer 13

and that only holds if the function is continuous which it is since it is a polynomial
:)

Answer 14

oh okay i get it now...that is by the IVT, what abt rolles thm?

Answer 15

that's by the IVT

Answer 16

rolles thm says that if f(a)=f(b), then f'(c)=0 where a < c < b

What this means is that if there are two points that have the same y value (or height) on the graph, then somewhere in between the two points will be a point that will have a tangent line of slope zero. Or put another way, in the middle of these two points will be a local min/max.


Now it turns out that if a function has a local min/max, then some translation of that function will have a different number of zeros. But, we can easily show that f(x) is one to one, which means that f(a)=f(b) implies a=b, but this violates one of the conditions in rolles thm. So what this means is that there are no two distinct points with the same y value (or height), which means that there are no local max/min points. So all this shows us that there is at most one zero.

Answer 17

now it all makes sense...
i wish i could give u and myininaya 100 medals..
Thanks alot:)

Answer 18

lalaly; can you help me with my questions?

Answer 19

hey nissan i got some question for u so answer tht and we can move on