With optimization, we are looking to determine a maximum or minimum value of a function over several variables. So, let's use an equation with an interesting curve in it:

If we take that curve and plot it to see if there is an apparent minimum value, we can use a script like the following that generates a plot as the result. (The %mathplotlib inline makes the plot appear inline of the Jupyter session, rather than creating the plot in a new window.)

%matplotlib inline
from scipy import optimize
import matplotlib.pyplot as plt
import numpy as np

def f(x):
    return x**4 - x**3 + x**2 + 1

x = np.linspace(-100, 50, 100)
plt.plot(x, f(x));  

Running this script in Jupyter, we see there is a natural minimum at x = 0.