Mara Cook

2022-06-24

II want to use the Intermediate Value Theorem and Rolle’s theorem to show that the graph of $f(x)={x}^{3}+2x+k$ crosses the x-axis exactly once, regardless of the value of the constant k.

I know I can use the intermediate value theorem, but I don't necessarily know how to show the change in a sign for two select inputs. any hero would be appreciated in that regard.

I also know the derivative of ${x}^{3}+2x+k$ is greater than zero, but what does that mean?

I know I can use the intermediate value theorem, but I don't necessarily know how to show the change in a sign for two select inputs. any hero would be appreciated in that regard.

I also know the derivative of ${x}^{3}+2x+k$ is greater than zero, but what does that mean?

benedictazk

Beginner2022-06-25Added 22 answers

Your function $f(x)={x}^{3}+2x+k$ has derivative $3{x}^{2}+2$ which, as you say, is greater than 0. This means that $f$ is strictly increasing. This results, for instance, from the mean value theorem. Now suppose $f$ is 0 at two distinct points $x$ and $y$, then $f(x)=f(y)$. But we must have either $xy$ or $yx$. We can assume $xy$ and then $f(x)f(y)$ since $f$ is increasing.

As you suggest, we use the intermediate value theorem to demonstrate that there is at least one solution to the problem. For any fixed $k$ we can choose $x$ large enough such that ${x}^{3}+2x+k0$. If we choose $x$ large but negative we get ${x}^{3}+2x+k0$. The intermediate value theorem now proves it.

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