Mathwords logoMathwords

Second Order Critical Point

Second Order Critical Point

A point on the graph of a function at which the second derivative is either 0 or undefined. A second order critical point may or may not be an inflection point.

Note: The phrase second order critical point is NOT in common usage among mathematicians or in textbooks. Nevertheless, it is a useful name for a type of point which otherwise has no name.

 

 

See also

Critical point

Key Formula

f′′(c)=0orf′′(c) is undefinedf''(c) = 0 \quad \text{or} \quad f''(c) \text{ is undefined}
Where:
  • f′′(c)f''(c) = The second derivative of the function f evaluated at x = c
  • cc = The x-coordinate of the second order critical point

Worked Example

Problem: Find all second order critical points of f(x) = x⁴ − 6x² + 8x + 5, and determine which are inflection points.
Step 1: Compute the first derivative.
f′(x)=4x3−12x+8f'(x) = 4x^3 - 12x + 8
Step 2: Compute the second derivative.
f′′(x)=12x2−12f''(x) = 12x^2 - 12
Step 3: Set the second derivative equal to zero and solve for x.
12x2−12=0  ⟹  x2=1  ⟹  x=−1 or x=112x^2 - 12 = 0 \implies x^2 = 1 \implies x = -1 \text{ or } x = 1
Step 4: Check whether the concavity changes at each point. For x = −1: f''(−2) = 12(4) − 12 = 36 > 0 and f''(0) = −12 < 0. The sign changes from positive to negative, so x = −1 is an inflection point.
f′′(−2)=36>0,f′′(0)=−12<0f''(-2) = 36 > 0, \quad f''(0) = -12 < 0
Step 5: For x = 1: f''(0) = −12 < 0 and f''(2) = 12(4) − 12 = 36 > 0. The sign changes from negative to positive, so x = 1 is also an inflection point.
f′′(0)=−12<0,f′′(2)=36>0f''(0) = -12 < 0, \quad f''(2) = 36 > 0
Answer: The second order critical points are at x = −1 and x = 1. Both are inflection points because the concavity changes sign at each.

Another Example

This example shows that a second order critical point is not automatically an inflection point. The second derivative is zero at x = 0, yet the function stays concave up, illustrating the key distinction.

Problem: Find the second order critical points of g(x) = x⁴ and determine whether each is an inflection point.
Step 1: Compute the first and second derivatives.
g′(x)=4x3,g′′(x)=12x2g'(x) = 4x^3, \quad g''(x) = 12x^2
Step 2: Set the second derivative equal to zero.
12x2=0  ⟹  x=012x^2 = 0 \implies x = 0
Step 3: Check the sign of g''(x) on both sides of x = 0. For x < 0, g''(x) = 12x² > 0. For x > 0, g''(x) = 12x² > 0. The second derivative is positive on both sides.
g′′(−1)=12>0,g′′(1)=12>0g''(-1) = 12 > 0, \quad g''(1) = 12 > 0
Step 4: Since the concavity does not change sign, x = 0 is a second order critical point but NOT an inflection point. The graph remains concave up on both sides.
Answer: x = 0 is a second order critical point of g(x) = x⁴, but it is not an inflection point because the concavity does not change.

Frequently Asked Questions

What is the difference between a critical point and a second order critical point?
A (first order) critical point occurs where the first derivative f'(x) is zero or undefined; these are candidates for local maxima and minima. A second order critical point occurs where the second derivative f''(x) is zero or undefined; these are candidates for inflection points. The two concepts involve different derivatives and identify different features of a graph.
Is every second order critical point an inflection point?
No. A second order critical point is only an inflection point if the concavity actually changes sign there. For example, f(x) = x⁴ has f''(0) = 0, but the graph is concave up on both sides of x = 0, so it is not an inflection point. You must always test the sign of the second derivative on each side.
When do you use second order critical points in calculus?
You use them when analyzing the concavity and shape of a curve. Finding where f''(x) = 0 or is undefined gives you candidate locations for inflection points. After confirming a sign change in f'', you can accurately describe where a graph changes from concave up to concave down or vice versa.

Second Order Critical Point vs. Critical Point (First Order)

Second Order Critical PointCritical Point (First Order)
DefinitionWhere f''(x) = 0 or f''(x) is undefinedWhere f'(x) = 0 or f'(x) is undefined
Derivative involvedSecond derivative f''(x)First derivative f'(x)
What it identifiesCandidate for an inflection pointCandidate for a local max or min
Confirmation testCheck if f'' changes sign across the pointUse first or second derivative test for extrema
Example: f(x) = x³x = 0 (since f''(0) = 0)x = 0 (since f'(0) = 0)

Why It Matters

In calculus, you routinely need to sketch curves and describe their shape. Second order critical points are the starting point for finding inflection points, which tell you where a graph switches between curving upward and curving downward. Understanding this concept helps you give a complete analysis of any function's behavior, which appears in AP Calculus curve-sketching problems and optimization contexts.

Common Mistakes

Mistake: Assuming every point where f''(x) = 0 is automatically an inflection point.
Correction: A second order critical point is only an inflection point if f'' actually changes sign. Always test the concavity on both sides. For instance, f(x) = x⁴ has f''(0) = 0 but no inflection point there.
Mistake: Forgetting to check where the second derivative is undefined, not just where it equals zero.
Correction: The second derivative can fail to exist at cusps, corners, or vertical tangents. These points are also second order critical points and must be checked for concavity changes. For example, f(x) = x^{1/3} has f''(0) undefined, and x = 0 is indeed an inflection point.

Related Terms