Answer to Question #252387 in Calculus for holi

Question

Find the distinct interval of length 1 containing a root or solution of f(x)=x ³ - 3x + 5 using IVT

Answer 1

Observe that

$$f(-3)=(-3)^3-3(-3)+5=-13<0$$

$$f(-1)=(-1)^3-3(-1)+5=7>0$$

Since $f$ is continuous, we may conclude by the IVT that $f$ has a root in $[-3, -1].$

Now

$$f(-2)=(-2)^3-3(-2)+5=3>0$$

So $f(-3)$ and $f(-2)$ are of opposite sign. Therefore, the IVT guarantees that $f$ has a root on $[-3, -2].$

We find the interval of length 1 containing a root or solution of $f(x)=x^3 - 3x + 5.$