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## The First and Second Derivatives and Local Minima

Whilst brushing up on linear algebra, I stumbled upon this webpage that provides a nice and short explanation on the first and second derivatives. To deepen my understanding I visualised the relationship between the first and second derivatives and local minimum / maximum.

### The Meaning of the First and Second Derivatives

- the function
- its first derivative is the slope of the tangent line at point x
- it tells whether the function is increasing or decreasing and how much it is increasing or decreasing
- if the first derivative is 0, x is called a critical point of

- its second derivative
- it tells if the first derivative is increasing or decreasing. when the second derivative is positive, the curve is concave up, and vice versa.
- when the second derivative is 0 then we do not know anything new about the behaviour of at that point

### Critical Points and the Second Derivative Test

- we can use the second derivative to find out when x is a local maximum or minimum
- suppose that x is a critical point (first derivative = 0) and the second derivative is positive.
- the second derivative tells us that the first derivative is increasing at that point and the graph is concave up.
- the only way to visualise this is local minimum where the slope of the function is zero but the graph is concave up.
- when the second derivative is negative with x being the critical point, it means that x is the local maximum.

### Visualisation with Python

Sympy is a great python library when playing with mathematical symbols and complex formulas. Taking a derivative is easy with Sympy.

```
import numpy as np
from sympy import *
import matplotlib.pyplot as plt
%matplotlib inline
```

Suppose

```
x = Symbol('x')
```

```
y = x**3 - 9*x**2 + 15*x - 7
```

```
## the first derivative
yfirst = y.diff(x)
print(yfirst)
```

3

x**2 - 18x + 15

```
## the second derivative
ysecond = yfirst.diff(x)
print(ysecond)
```

6*x - 18

```
## input x range
test_x = np.linspace(-1, 7, 50)
```

```
## corresponding y, first_derivative, second_derivative
test_y = [y.subs({x:v}) for v in test_x]
test_y_f = [yfirst.subs({x:v}) for v in test_x]
test_y_s = [ysecond.subs({x:v}) for v in test_x]
```

```
fig, ax = plt.subplots(figsize=(10, 10))
ax.plot(test_x, test_y, color='black', label='function_f')
ax.plot(test_x, test_y_f, color='red', label='first_order')
ax.plot(test_x, test_y_s, color='blue', label='second_order')
ax.plot(test_x, np.zeros(len(test_x)), 'g--', label='y=0')
circle1 = plt.Circle((1, 0), 1, color='red', fill=False)
circle2 = plt.Circle((5, -32), 1, color='r', fill=False)
ax.add_artist(circle1)
ax.add_artist(circle2)
plt.annotate('local maximum', xy=(1, 0), xytext=(1, 2.5), fontsize=15)
plt.annotate('local minimum', xy=(5, -32), xytext=(5, -30), fontsize=15)
ax.legend(fontsize=15)
plt.show()
```

As stated above, the function is at local maximum when the first derivative is at 0 and the second derivative is negative. And itâ€™s at local minimum when the second derivative is positive.