Critical Points

Nerd Cafe

Mathematical Background

• Critical points occur where the derivative of a function equals zero or is undefined.

• These points help identify where the function might have local maxima, minima, or points of inflection.

Example 1:

$$f(x)=x^{2}$$

• Derivative:

$$f\acute{}(x)=2x$$

• Set derivative to zero:

$$2x=0\Rightarrow x=0$$

• Critical Point:

$$x=0$$

• Python Code:

from sympy import symbols, diff, solve

x = symbols('x')
f = x**2
f_prime = diff(f, x)
critical_points = solve(f_prime, x)
print("Critical Points:", critical_points)

• Output:

Critical Points: [0]

Example 2:

$$f(x)=x^{3}-3x$$

• Derivative:

$$f\acute{}(x)=3x^{2}-3$$

• Solve:

$$3x^{2}-3=0\Rightarrow x=\pm 1$$

• Python Code:

from sympy import symbols, diff, solve

x = symbols('x')
f = x**3 - 3*x
f_prime = diff(f, x)
critical_points = solve(f_prime, x)
print("Critical Points:", critical_points)

• Output:

Critical Points: [-1, 1]

Example 3:

$$f(x)=\frac{1}{x}$$

• Derivative:

$$f\acute{}(x)=-\frac{1}{x^{2}}$$

This is never zero, but undefined at x=0.

• Python Code:

from sympy import symbols, diff, solve

x = symbols('x')
f = 1/x
f_prime = diff(f, x)
critical_points = solve(f_prime, x)
print("Critical Points (zero derivative):", critical_points)
# Check undefined
print("Derivative undefined at x=0?", f_prime.subs(x, 0) if x != 0 else "Undefined")

• Output:

Critical Points (zero derivative): []
Derivative undefined at x=0? zoo

Example 4:

$$f(x)=x^{4}-4x^{2}$$

• Derivative:

$$f\acute{}(x)=4x^{3}-8x=4x\left( x^{2}-2 \right)\Rightarrow x=0,x=\pm \sqrt{2}$$

• Python Code:

from sympy import symbols, diff, solve

x = symbols('x')
f = x**4 - 4*x**2
f_prime = diff(f, x)
critical_points = solve(f_prime, x)
print("Critical Points:", critical_points)

• Output:

Critical Points: [0, -sqrt(2), sqrt(2)]

Example 5:

$$f(x)=\sqrt{x}$$

• Derivative:

$$f\acute{}(x)=\frac{1}{2\sqrt{x}}$$

• Python Code:

from sympy import symbols, diff, solve, sqrt

x = symbols('x')
f = sqrt(x)
f_prime = diff(f, x)
# Solve f'(x) = 0
critical_points = solve(f_prime, x)
print("Critical Points:", critical_points)
# Note undefined at x=0
print("Derivative undefined at x=0?", f_prime.subs(x, 0))

• Output:

Critical Points: []
Derivative undefined at x=0? zoo

Example 6:

$$f(x)=x.e^{x}$$

• Derivative:

$$f\acute{}(x)=e^{x}+x.e^{x}=e^{x}\left( 1+x \right)\Rightarrow \left( 1+x \right)=0\Rightarrow x=-1$$

• Python Code:

from sympy import symbols, diff, solve, exp

x = symbols('x')
f = x * exp(x)
f_prime = diff(f, x)
critical_points = solve(f_prime, x)
print("Critical Points:", critical_points)

• Output:

Critical Points: [-1]

Example 7:

$$f(x)=ln\left( x \right)$$

• Derivative:

$$f\acute{}(x)=\frac{1}{x}\Rightarrow never \: zero, \: undefined \: \: at \: \: x=0$$

• Python Code:

from sympy import symbols, diff, solve, ln

x = symbols('x')
f = ln(x)
f_prime = diff(f, x)
critical_points = solve(f_prime, x)
print("Critical Points:", critical_points)

• Output:

Critical Points: []

Example 8:

$$f(x)=sin\left( x \right)$$

• Derivative:

$$f\acute{}(x)=cos\left( x \right)\Rightarrow cos\left( x \right)=0\Rightarrow x=\frac{\pi}{2}+n\pi$$

• Python Code:

from sympy import symbols, sin, pi, diff, solve, solveset, Interval

x = symbols('x')
f = sin(x)
f_prime = diff(f, x)
critical_points = solve(f_prime, x)
print("Critical Points (symbolic):", critical_points)  # May return [pi/2 + n*pi] if general solution used

# For specific interval:
print("Critical Points in [0, 2π]:", solveset(f_prime, x, domain=Interval(0, 2*pi)))

• Output:

Critical Points (symbolic): [pi/2, 3*pi/2]
Critical Points in [0, 2π]: {pi/2, 3*pi/2}

Keywords

critical points, calculus, derivatives, local maxima, local minima, inflection points, turning points, first derivative, second derivative, optimization, increasing function, decreasing function, concavity, stationary points, curve sketching, derivative test, extrema, function analysis, symbolic computation, Python SymPy, nerd cafe