Question

Using the Intermediate Value Theorem and a calculator, find an interval of length 0.01 that contains a root of x5−x2+2x+3=0, rounding off interval endpoints to the nearest hundredth.

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Answer 1

$$Solution$$

For $f(x) = x^5 - x^2 + 2x + 3$;

$f(0) = 3\\ f(-1) = -1 - 1 - 2 + 3 = -1$

Thus, as this is a continuous function, from the Intermediate Value Theorem, we know that, as $f(-1) < 0$ and $f(0) > 0$, there is a root between $-1$ and $0.$

While we could use the bisection method, the easiest thing is to find $f(x)$ for $x$ between $-1$ and in increments of $.01$

$f(-0.88) = -0.0621319168000003\\ f(-0.87) = 0.00467907929999978$

Then, the Intermediate Value Theorem tells us, at $f(-0.88) < 0$ and $f(-0.87) = 0.00467907929999978$, there is a root in the interval $(-0.88, -0.87)$