Critical Point

A critical point is a point on a function where the derivative is 0.

Description

Critical points are members of a function's domain. For a given f, c can only be a critical point if f(c) exists.

For a differentiable function, critical points are defined by the first derivative being equal to 0. The tangent of the line at these points is horizontal. Such a point must be one of...

• a local minimum
• a local maximum
• an inflection point

Fermat's theorem or the interior extremum theorem states that all minima and maxima of a differentiable function are critical points. The second derivative can then be used to discriminate between these extrema and any inflection points that also have a first derivative of 0. Following this method, critical points are an important tool for optimization problems.

If a function is not everywhere differentiable, note that any c where f(c) exists but f'(c) does not exist is also a critical point. For example, f(x)=|x| is defined for all x but is not differentiable at x=0. In this case, x=0 is the unique critical point.

In the multivariate and everywhere differentiable case, a critical point is defined by all partial derivatives being equal to 0. The tangent plane at these points is horizontal. Such a point must be one of...

• a local minimum
• a local maximum
• a saddle point

Univariate Critical Points

Identification

Critical points can be found by taking the first derivative and setting it equal to 0.

If the first derivative is a polynomial like 30x⁴ + 132x³ - 90x², it can be factored into (6x²)(5x−3)(x+5), and the roots of this function are clearly at x=−5, x=0, and x=3/5. If a polynomial cannot be factored and is in the form ax² + bx + c, use the quadratic formula to identify the roots:

Remember that there are two solutions!

If the first derivative is a ratio, first identify the roots of the denominator. These are necessarily important points where the first derivative is undefined due to division by 0. They are then also critical points if the original function is defined there. The second step is to identify the roots of the numerator.

Sinusoidal functions have infinitely many critical points, and are generally expressed like: a + 2πn for all n.

Classification

Evaluate the second derivative at all critical points to reveal concavity.

• If positive, then the line is concave up at the point, and the point is a local minimum.
• If negative, then the line is concave down at the point, and the point is a local maximum.
• If equal to 0, then the classification is indeterminate.

As an example of the third case, consider x⁴, -x⁴, and x³. Clearly each has a critical point at 0, and evaluating each of their second derivatives at 0 gives 0. The first is the global minimum; the second is the global maximum; and the third is just an inflection point.

Bivariate Critical Points

Identification

Take the partial derivatives of a function and set them equal to 0. This creates a system of equations that can be solved for critical points (i.e., pairs of x=x_c and y=y_c).

Classification

Quadratic Case

If a line is quadratic, i.e. of the form w = ax² + bxy + cy², then there is a special approach to classifying critical points. Clearly the origin is a critical point. If furthermore a is non-zero, then the equation can be rewritten like:

ax² + bxy + cy²

a(x² + (b/a)xy) + cy²

a(x+(b/2a)y)² + (c - (b²/4a))y²

(1/4a) [4a²(x+(b/2a)y)² + (4ac - b²)y²]

The importance is that this reformulation is a sum of two squares where the first term (i.e., 4a²(x+(b/2a)y)²) is always positive. These rules for classifying the origin follow:

1. If 4ac - b²>0, then the sum of two squares is always positive. The sign of the overall reformulation can only vary with a:

   1. If a>0, then the plane is concave up at the origin, and it is a local minimum.

   2. If a<0, then the plane is concave down at the origin, and it is a local maximum.

2. If 4ac - b²<0, then the second term is oppositely signed and the origin is a saddle point.

3. If 4ac - b²=0, then the origin is degenerate. There is a line of infinitely many critical points passing through the origin. The plane varies in one dimension but not the other. More information, e.g. higher order derivatives, is needed to classify this case; for now the classification is indeterminate.

General Case

Calculate the second derivatives, which are then denoted as:

• f_xx = ∂²f/∂x²

• f_xy = ∂²f/∂x∂y

  • Note that ∂²f/∂x∂y = ∂²f/∂y∂x; order does not matter.

• f_yy = ∂²f/∂y²

Evaluate these at each critical point:

• A = f_xx(x_c,y_c)

• B = f_xy(x_c,y_c)

• C = f_yy(x_c,y_c)

These rules for classifying critical points follow:

1. If AC-B²>0...

   1. If A>0, then the plane is concave up at the point, and the critical point is a local minimum.

   2. If A<0, then the plane is concave down at the point, and the point is a local maximum.

2. If AC-B²<0, then the point is a saddle point.

3. If AC-B²=0, then the point is degenerate. The the classification is indeterminate.

This all follows from a Taylor expansion:

Δf ≈ (∂f/∂x + ∂f/∂y) + ( (1/2)(∂²f/∂x²) + ∂²f/∂x∂y + (1/2)(∂²f/∂y²) ) ...

But obviously the first partial derivatives are 0 at a critical point, so:

Δf ≈ (1/2)(∂²f/∂x²) + ∂²f/∂x∂y + (1/2)(∂²f/∂y²) ...

And so the general case (unless the critical point is degenerate!) is approximated by a quadratic formula, and the quadratic approach can be used like:

• a = (1/2)(∂²f/∂x²) = (1/2)A

• b = (1/2)(∂²f/∂x∂y) = B

• c = (1/2)(∂²f/∂y²) = (1/2)C

Proving quadratic case using general case

Taking the special case formulation as w, note that:

• A = w_xx(x_c,y_c) = 2a

• B = w_xy(x_c,y_c) = b

• C = w_yy(x_c,y_c) = 2c

AC - B²

(2a)(2c) - (b)²

4ac - b²

Therefore the rules for classifying critical points are equivalent between the two cases.

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