Suppose \(x_{t}\) is a time series with a linear and a quadratic trend, with white noise superimposed:

\[ x_{t}=\beta_{0}+\beta_{1} t+\beta_{2} t^{2}+w_{t}, \quad \text { for } t=1,2, \ldots \]

where \(\beta_{0}, \beta_{1}\), and \(\beta_{2}\) are constants, and \(w_{t}\) is a white noise with variance \(\sigma^{2}\).

(a) Is \(\left\{x_{t}\right\}\) (weakly) stationary? Why or why not?

(b) What is the ACF of \(\left\{x_{t}\right\}\) ?

(c) Simulate 200 observations for \(t=1,2, \ldots, 500\) following the \(\left\{x_{t}\right\}\) with \(\beta_{0}=1, \beta_{1}=0.1\), and \(\beta_{2}=0.01\), and compute the sample ACF.

Plot the series and the ACF.