Another solution using essentially a dilate operator:

import numpy as np
from scipy.ndimage import rank_filter

def find_local_maxima(x):
   x_dilate = rank_filter(x, -1, size=3)
   return x_dilate == x

and for the minima:

def find_local_minima(x):
   x_erode = rank_filter(x, -0, size=3)
   return x_erode == x

Another solution using essentially a dilate operator:

import numpy as np
from scipy.ndimage import rank_filter

def find_local_maxima(x):
   x_dilate = rank_filter(x, -1, size=3)
   return x_dilate == x

and for the minima:

def find_local_minima(x):
   x_erode = rank_filter(x, -0, size=3)
   return x_erode == x

Also, from scipy.ndimage you can replace rank_filter(x, -1, size=3) with grey_dilation and rank_filter(x, 0, size=3) with grey_erosion. This won't require a local sort, so it is slightly faster.