Where will a prediction interval or a confidence interval be narrower: near the mean or further from the mean?

Both prediction and confidence intervals are narrower near the mean, this can be easily seen in formula of corresponding margin of errors.

Following is the margin of error of confidence interval.
#E = t_{\alpha/2, df=n-2} \times s_e \sqrt{ ( \frac{1}{n } + \frac{ (x_0 - \bar{x})^2}{S_{x x} } ) }#

Following is the margin of error for prediction interval

#E = t_{\alpha/2, df=n-2} \times s_e \sqrt{ (1 + \frac{1}{n } + \frac{ (x_0 - \bar{x})^2}{S_{x x} } ) }#

In both of these, we see the term #(x_0 - \bar{x})^2#, which scales as the square of the distance of the prediction point from mean. This is why CI and PI is narrowest at the mean.