2021-12-17

How do I use the intermediate value theorem to determine whether ${x}^{5}+3{x}^{2}-1=0$ has a solution over the interval $\left[0,3\right]?$

Explanation:
Bolzanos

Joseph Lewis

Explanation:
According to the intermediate value theorem:
If the function $f\left(x\right)$ is continuous on the interval $\left[a,b\right]$ and $y$ is a number between , then there exists a number $x=c$ in the open interval $\left(a,b\right)$ such that $y=f\left(c\right)$
Here,
$f\left(x\right)={x}^{5}+3{x}^{2}-1$
is a continouos function on the interval $\left[0,3\right]$ as $f\left(x\right)$ is a polynomial function.
$f\left(0\right)=-1$
$f\left(3\right)=243+27-1=269$
$0\in \left(f\left(0\right),f\left(3\right)\right)$
As $f\left(x\right)$ changes sign from $f\left(0\right)$ as $f\left(3\right)$
$\mathrm{\exists }c\in \left(0,3\right)$ such that $f\left(c\right)=0$
The intermediate value theorem is used in this situation.

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