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the question I am trying to solve is the following:

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?

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1 Answer

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$$f(x)=x^5−x^2+2x+3$$

As you can see $f(0)=3>0$ and $f(-1)=-1<0$

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

Now calculate the value of $$f(-\frac{1}{2})=\frac{55}{32}>0$$

Thus now our interval is shortened and it is $(-1,-\frac{1}{2})$

$$f(-\frac{3}{4})=\frac{717}{1024}>0$$

Our interval is now $(-1,-\frac{3}{4})$

$$f(-\frac{7}{8})=-\frac{935}{32768}<0$$

Our interval is now $(-\frac{7}{8},-\frac{3}{4})$

similarly, keep doing until you get the desired result

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