Misusing the second-derivative test

The second-derivative test decides whether a stationary point is a maximum or a minimum. The rule is: at a point where $\frac{dy}{dx}=0$, if $\frac{d^{2}y}{dx^{2}}<0$ the point is a maximum, and if $\frac{d^{2}y}{dx^{2}}>0$ it is a minimum.

Two misuses are common. The first is reversing the convention, reading a positive second derivative as a maximum, because 'positive' feels like 'the top'.

The second is applying the test at a point that is not stationary at all, so the sign of $\frac{d^{2}y}{dx^{2}}$ there decides nothing about a turning point. Both are tempting under time pressure, when the routine gets run from memory without checking which value goes with which conclusion.

For $y=x^{2}-6x+5$, $\frac{dy}{dx}=2x-6=0$ gives the stationary point at $x=3$. The second derivative there is:

The reversed convention draws the wrong conclusion: Wrong,'the second derivative is positive, so the point is a maximum.'

Correct, a positive second derivative means the curve is concave up, shaped like a valley, so the stationary point at $x=3$ is a minimum. A negative second derivative would mean concave down, a hill, and hence a maximum.

1. 1

   Find the stationary points first

   Solve $\frac{dy}{dx}=0$. The test only decides the nature of these points, nowhere else.

2. 2

   Differentiate again

   Find $\frac{d^{2}y}{dx^{2}}$ as a function of $x$.

3. 3

   Substitute the stationary x-value

   Put each stationary $x$ into $\frac{d^{2}y}{dx^{2}}$ and read its sign.

4. 4

   Apply the convention

   $\frac{d^{2}y}{dx^{2}}>0\Rightarrow$ minimum; $\frac{d^{2}y}{dx^{2}}<0\Rightarrow$ maximum.

5. 5

   Handle the zero case

   If $\frac{d^{2}y}{dx^{2}}=0$, the test is inconclusive; check the sign of $\frac{dy}{dx}$ just before and just after the point instead.

The fix is not more theory but a locked routine, and that is what our teachers build. We drill 'stationary first, sign second': solve $\frac{dy}{dx}=0$, then read the sign of $\frac{d^{2}y}{dx^{2}}$ only at those $x$-values, and say 'positive smiles, minimum' aloud each time until it sticks.

In a one-to-one lesson we can catch the exact moment the convention flips and re-anchor it with a quick sketch of a valley and a hill. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

Message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.