Identifying Peaks (Local Maxima) and Troughs (Local Minima) in Data and Labeling them

I have data that resembles this:

import pandas as pd
import random

random.seed(901)
rand_list = []
for i in range(20):
    x = random.randint(480, 600)
    rand_list.append(x/10)

df = pd.DataFrame({'INDEX':rand_list})
df['INDEX_DIFF'] = df.INDEX.diff()
df

Output:

      INDEX   INDEX_DIFF
 0    53.5    NaN
 1    56.4    2.9
 2    51.7    -4.7
 3    49.4    -2.3
 4    55.4    6.0
 5    49.9    -5.5
 6    52.9    3.0
 7    57.7    4.8
 8    54.2    -3.5
 9    51.4    -2.8
10    57.1    5.7
11    56.7    -0.4
12    58.5    1.8
13    52.1    -6.4
14    57.6    5.5
15    56.1    -1.5
16    54.2    -1.9
17    52.9    -1.3
18    56.6    3.7
19    53.2    -3.4

I want to identify instances where the INDEX value crosses 50 from below and keeps growing while above 50. The first instance when INDEX_DIFF shows -ve indicates that the INDEX has peaked above 50.

For instance, the 6th value 49.9 crosses 50 and continues to grow (52.9, 57.7) before peaking at 57.7 and dropping to 54.2 in the next instance. The INDEX_DIFF gives the first negative value since crossing 50 from below.

How can I check this condition and label this as a Peak in a new column in the pandas dataframe?

I have tried to use Last occurrence of a condition, but I don't know how to check this elaborate condition.

You could use masks to identify the chunks above 50 (and that started below 50), then get the groupby.idxmax:

m1 = df['INDEX'].shift().lt(50)
m2 = df['INDEX'].gt(50)

group = (m1 & m2).cumsum()

maxima = df[group>0].groupby(group)['INDEX'].idxmax().tolist()
# [4, 12]

df.loc[maxima, 'peak'] = 'X'

Output:

    INDEX  INDEX_DIFF peak
0    53.5         NaN  NaN
1    56.4         2.9  NaN
2    51.7        -4.7  NaN
3    49.4        -2.3  NaN
4    55.4         6.0    X
5    49.9        -5.5  NaN
6    52.9         3.0  NaN
7    57.7         4.8  NaN
8    54.2        -3.5  NaN
9    51.4        -2.8  NaN
10   57.1         5.7  NaN
11   56.7        -0.4  NaN
12   58.5         1.8    X
13   52.1        -6.4  NaN
14   57.6         5.5  NaN
15   56.1        -1.5  NaN
16   54.2        -1.9  NaN
17   52.9        -1.3  NaN
18   56.6         3.7  NaN
19   53.2        -3.4  NaN

If you want all peaks (not just the largest one):

m1 = df['INDEX'].shift().lt(50)
m2 = df['INDEX'].gt(50)
m3 = (m1 & m2).cummax()

peak = df['INDEX'].diff().gt(0) & df['INDEX'].diff(-1).gt(0)

df['peak'] = np.where(m3 & peak, 'X', '')

Output:

    INDEX  INDEX_DIFF peak
0    53.5         NaN
1    56.4         2.9
2    51.7        -4.7
3    49.4        -2.3
4    55.4         6.0    X
5    49.9        -5.5
6    52.9         3.0
7    57.7         4.8    X
8    54.2        -3.5
9    51.4        -2.8
10   57.1         5.7    X
11   56.7        -0.4
12   58.5         1.8    X
13   52.1        -6.4
14   57.6         5.5    X
15   56.1        -1.5
16   54.2        -1.9
17   52.9        -1.3
18   56.6         3.7    X
19   53.2        -3.4

Graph approach 1:

Graph approach 2: