Second-Order Condition¶

Proposition¶

Let $a < b$ be real numbers, and let $f \colon (a, b) \to \mathbb{R}$ be a function which is twice-differentiable at its stationary point $x_0\in(a,b)\text{.}$ Then $x_0$ is a local maximum point of $f$ if $f''(x_0)<0\text{,}$ and it is a local minimum point of $f$ if $f''(x_0)>0\text{.}$

Interpretation¶

The second-order condition decides what kind of stationary point $x_0$ is from the sign of $f''(x_0)$ . At a stationary point the tangent is horizontal; $f''(x_0)<0$ means that the graph bends downwards there, giving a local maximum, and $f''(x_0)>0$ means that it bends upwards, giving a local minimum.

Three panels: a downward parabola with a local maximum where f'' is negative, an upward parabola with a local minimum where f'' is positive, and x cubed at 0, where f'' = 0 gives no conclusion.

The left and middle panels show the two decisive cases: a downward parabola with a local maximum and an upward one with a local minimum. In the right panel, $x^3$ has $f'(0)=f''(0)=0$ and no extremum at $0$ ; the function $x^4$ has the same values $f'(0)=f''(0)=0$ and a minimum. So $f''(x_0)=0$ gives no conclusion either way.

The figure marks every stationary point and its sign of $f''$ . The example $2x^4-\frac13x^6$ has $f''(0)=0$ $0$ , so the condition is silent there, as are $x^3$ and $x^4$ at $0$ , one without and one with an extremum.

The reason lies in the sign of $f'$ near $x_0$ . Since $f'(x_0)=0$ $0$ , the difference quotient of $f'$ is $f'(x)/(x-x_0)$ $x_{0})$ , and it tends to $f''(x_0)$ . If $f''(x_0)>0$ $0$ , then close to $x_0$ we have $f'<0$ on the left and $f'>0$ on the right, so $f$ decreases and then increases: a local minimum. The condition only applies at stationary points, found with the first-order condition; a positive second derivative at a point where $f'\neq0$ says nothing about extrema.

The conclusion is local. If $f''>0$ on the whole interval, $f$ is strictly convex, and a stationary point is the global minimum. A common slip is to read the condition backwards: a local minimum does not force $f''(x_0)>0$ $0$ , as $x^4$ shows; what does hold at a local minimum of a twice differentiable function is $f''(x_0)\ge0$ $0$ . Math IV extends the test to functions of several variables in the conditions for unconstrained extrema.

Key Results¶

A negative second derivative indicates local maximality.
A positive second derivative indicates local minimality.
A zero second derivative indicates that further analysis is needed.
The condition is sufficient but not necessary.

Examples¶

The function $f(x)=2x^4-\frac{1}{3}x^6$ has three stationary points $x=0\text{,}$ $x=2\text{,}$ and $x=-2\text{.}$ We plug $x=0\text{,}$ $2\text{,}$ and $-2$ in $f''(x)=24x^2-10x^4$ to find $f''(2)=f''(-2)=-64<0$ and $f''(0)=0\text{.}$ Then, we conclude that $x=2$ and $x=-2$ are local maximum points. The condition gives no conclusion at $x=0\text{,}$ but directly $f(x)=x^4\left(2-\frac{1}{3}x^2\right)\ge0=f(0)$ for $|x|\le\sqrt{6}\text{,}$ so $x=0$ is a local minimum point.

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 9

Aliases: SOC, second order condition, second-order sufficient condition Keywords: second derivative test Symbols: element of, real numbers, square root

Last updated: 20 October 2026 03:02