Maxima, Minima & the 2nd-Derivative Test

Last lesson left you with a list of suspects — the flat spots where f'(x) = 0. The question now is: peak or trough? You could sample f' on both sides and watch the sign flip, but there’s a shortcut. Just look at the curvature at the point.

That’s all the second derivative is. f'' tells you whether the curve is bending up like a bowl or down like a dome. A bowl with a flat bottom is a minimum; a dome with a flat top is a maximum. One sign, one answer — and it’s the calculus idea GATE asks most often.

The same curvature check scales up: in ML the multivariable version (the Hessian) is how Newton’s method and convexity tests decide whether a flat point is a usable minimum, so this sign-of-f'' instinct is one you’ll lean on for real models too.

The second-derivative test

At a critical point x* where f'(x*) = 0:

f''(x*) > 0   ⇒   concave up (bowl)   ⇒   LOCAL MINIMUM
f''(x*) < 0   ⇒   concave down (dome)  ⇒   LOCAL MAXIMUM
f''(x*) = 0   ⇒   INCONCLUSIVE  (could be min, max, or neither)

The first two lines are the workhorses. The third is the trap, covered below.

Try it on a cubic. Switch the function below to x³ − x and slide the point to its two critical spots — at x ≈ −0.58 the tangent flattens at the top of a dome (f'' < 0, a local max), and at x ≈ +0.58 it flattens at the bottom of a bowl (f'' > 0, a local min). Same shape as the x³ − 3x example below, same verdict.

Local vs global

The test certifies a local extremum — the highest or lowest point in a small neighbourhood. It says nothing about the function as a whole. A valley can sit above a deeper valley elsewhere; a hilltop can be dwarfed by a taller hill. A global (or absolute) extremum is the largest or smallest value over the whole domain. So f''(x*) > 0 guarantees a local min — never automatically a global one. (Finding global extrema needs the closed-interval method of the next lesson.)

How GATE asks this

The classic is a one-line MCQ/NAT: “if f'(x*) = 0 and f''(x*) > 0, then f has a ___ at x*” — and the tempting wrong answer is global minimum. Or it hands you a cubic and asks you to classify its critical points. The drill: solve f'(x) = 0 for the candidates, then plug each into f'' and read the sign.

Worked example — a real GATE DA 2024 question

f is twice differentiable, f'(x*) = 0, and f''(x*) > 0. What can you conclude about f at x*?

This is GATE DA 2024. By the second-derivative test, a flat point with positive curvature is concave up — a bowl — so:

f'(x*) = 0   and   f''(x*) > 0    ⇒    f has a LOCAL MINIMUM at x*

The trap answer is global minimum. Positive curvature is a purely local statement: it describes the bowl right around x* and says nothing about how low f might dip elsewhere. The correct GATE answer is local minimum.

Now apply the same test to classify a concrete cubic — the x³ − 3x from the last lesson, this time with the second derivative instead of sign analysis:

f(x)  = x³ − 3x
f'(x) = 3x² − 3 = 0      →   x = −1  and  x = +1     (the critical points)
f''(x) = 6x

f''(−1) = 6(−1) = −6  < 0    ⇒  LOCAL MAXIMUM at x = −1   (value f(−1) = 2)
f''(+1) = 6(+1) = +6  > 0    ⇒  LOCAL MINIMUM at x = +1   (value f(1) = −2)

One substitution per critical point settles it — no sign table needed. (Both agree with the first-derivative analysis from the previous lesson.)

Quick check