Answer
The graph of function $f$ is shown below.
The solution to equation $f(x)=0$ is $x\approx-1.7693$.
Work Step by Step
$$f(\theta)=\theta^3-2\theta+2$$
a) Domain: $f(\theta)$ is defined on $R$.
- Continuity: For all $\theta=c\in R$, we always have $$\lim_{\theta\to c}f(\theta)=c^3-2c+2=f(c)$$
Therefore, $f(\theta)$ is continuous on $R$.
- At $\theta=-2$: $f(-2)=(-2)^3-2\times(-2)+2=-8+4+2=-2\lt0$
- At $\theta=0$: $f(0)=0^3-2\times0+2=0-0+2=2\gt0$
So $f(\theta)$ has changed its sign as $\theta$ changes from $-2$ to $0$.
Since $f$ is continuous on $R$, according to the Intermediate Value Theorem, there is a value of $\theta=c\in[-2,0]$ for which $f(c)=0$. In other words, $f$ has a zero between $-2$ and $0$.
b) The graph of function $f(\theta)$ is shown below.
As noted in the graph, $f$ passes the $x$-axis at point $(-1.7693, 0)$. So the solution to the equation $f(x)=0$ is $x\approx-1.7693$.
c) Evaluate the exact answer: $$\Big(\sqrt{\frac{19}{27}}-1\Big)^{1/3}-\Big(\sqrt{\frac{19}{27}}+1\Big)^{1/3}\approx-1.769292$$
The exact value here is close to the value found in part b).
