maduregimc

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 $[0,3]?$

Nadine Salcido

Beginner2021-12-18Added 34 answers

Explanation:

Bolzanos

Bolzanos

Joseph Lewis

Beginner2021-12-19Added 43 answers

Explanation:

According to the intermediate value theorem:

If the function $f\left(x\right)$ is continuous on the interval $[a,b]$ and $y$ is a number between $f\left(a\right)\text{}\text{and}\text{}f\left(b\right)$, then there exists a number $x=c$ in the open interval $(a,b)$ 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 $[0,3]$ as $f\left(x\right)$ is a polynomial function.

$f\left(0\right)=-1$

$f\left(3\right)=243+27-1=269$

$0\in (f\left(0\right),f\left(3\right))$

As $f\left(x\right)$ changes sign from $f\left(0\right)$ as $f\left(3\right)$

$\mathrm{\exists}c\in (0,3)$ such that $f\left(c\right)=0$

The intermediate value theorem is used in this situation.

RizerMix

Skilled2021-12-29Added 437 answers

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