Semaj Christian

2022-06-21

Using the Intermediate Value Theorem and a calculator, find an interval of length 0.01 that contains a root of ${x}^{5}-{x}^{2}+2x+3=0$, rounding off interval endpoints to the nearest hundredth.

I've done a few things like entering values into the given equation until I get two values who are 0.01 apart and results are negative and positive ($-1.15$ & $-1.16$), but these answers were incorrect.

I'm at the point where I'm thinking there is not enough information to solve. Any ideas?

humbast2

$f\left(x\right)={x}^{5}-{x}^{2}+2x+3$
As you can see $f\left(0\right)=3>0$ and $f\left(-1\right)=-1<0$

Thus there is at least one root of $f\left(x\right)=0$ in Interval $\left(-1,0\right)$

Now calculate the value of
$f\left(-\frac{1}{2}\right)=\frac{55}{32}>0$
Thus now our interval is shortened and it is $\left(-1,-\frac{1}{2}\right)$
$f\left(-\frac{3}{4}\right)=\frac{717}{1024}>0$
Our interval is now $\left(-1,-\frac{3}{4}\right)$
$f\left(-\frac{7}{8}\right)=-\frac{935}{32768}<0$
Our interval is now $\left(-\frac{7}{8},-\frac{3}{4}\right)$

similarly, keep doing until you get the desired result

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