kevinnn said:Homework Statement
show that the equation x^3-15x+C=0 has at most one root on the interval [-2,2]
Homework Equations
The Attempt at a Solution
I know I need to use Rolle's theorem but I'm not sure how to find the answer. Thanks.
kevinnn said:Ohhh well if it had two roots then the function would have both positive and negative values on the interval.
kevinnn said:No sorry. It tells me that a number c exists in the interval such that f'(c)=0
kevinnn said:Well if f(a)=f(b)=0 then the function is a constant over [a,b] so there are infinitely many points where the derivative equals zero. The opposite of what I was trying to show. So what am I missing.
kevinnn said:I did that and when I take the derivative and set it equal to zero I get plus/minus the square root of 5. Which is not even the interval. That would tell me that there are not at most one, but no roots in [-2,2].
kevinnn said:I did that and when I take the derivative and set it equal to zero I get plus/minus the square root of 5. Which is not even the interval. That would tell me that there are not at most one, but no roots in [-2,2].
LCKurtz said:No roots of what?
kevinnn said:Ohhhhh now I got you. So since the maximum and minimum of the graph are at plus minus the square root of five that means that the graph can only be either increasing or decreasing on our interval so the most the graph can cross the x-axis is once. Thanks.
Rolle's theorem is a mathematical theorem that states that if a continuous function has the same values at two points, and is differentiable between those two points, then there exists at least one point between them where the derivative of the function is equal to zero. This point is known as a "root" of the function, where the function crosses the x-axis. Rolle's theorem is helpful in proving that there is at most one root for a given function.
Rolle's theorem is crucial in proving the existence of at most one root because it provides a mathematical framework for finding the exact location of the root. Without this theorem, it would be difficult to determine where the root is located and whether there are multiple roots.
Yes, Rolle's theorem can be used to prove that a function has no roots. If a function is continuous and differentiable on a given interval, and the derivative of the function is never equal to zero on that interval, then there are no roots on that interval. This is because the function never crosses the x-axis, which is a necessary condition for a root to exist.
One limitation of using Rolle's theorem is that it only applies to continuous and differentiable functions. If a function is not continuous or differentiable on a given interval, then Rolle's theorem cannot be used to prove the existence of at most one root. Additionally, Rolle's theorem only guarantees the existence of one root, but it does not necessarily provide a method for finding the root.
No, Rolle's theorem can only be used to prove the existence of at most one root. If a function has multiple roots, then Rolle's theorem cannot be used to determine their locations. In this case, other methods such as the intermediate value theorem or the mean value theorem may be used to prove the existence of multiple roots.