Class 12 Statistics Notes · GSEB

Second Order Derivative

Second Order Derivative — learn f'(x), concavity, and the second derivative test for maxima and minima. GSEB Class 12 Commerce Statistics notes with an interactive concavity explorer.

Last updated: 24 Sep 2026

Notes

What Is a Second Order Derivative?

The derivative of the first order derivative — the rate of change of the rate of change. It is denoted by f″(x) or d²y/dx².

★ Second Order Derivative

f''(x) = \frac{d}{dx}[f'(x)] = \frac{d^2 y}{dx^2}

If f′(x) is the speedometer, f″(x) is the accelerator pedal: it tells you whether the rate itself is rising or falling. Its biggest exam use is the maximum–minimum test.

f″(x)

prime-prime

d²y/dx²

Leibniz form

dot form

f⁽²⁾(x)

superscript form

It is a derivative of a derivative

To get f″ you differentiate f′ — you do not square f′. For y = x³: y′ = 3x², then y″ = 6x (not 9x²).

Concavity: What f″ Tells You Geometrically

The sign of f″ decides which way the curve bends. Drag the point along y = x³ − 3x and watch the colour change.

Concavity Explorer — y = x³ − 3x (f″ = 6x)

Move point P — x = 1.20

f′(x) = 3x² − 3

1.32

f″(x) = 6x

7.20

Concavity

Bowl up (∪)

Read the colour flip

Cross x = 0: red (f″ < 0, bowl down) becomes green (f″ > 0, bowl up). The point where concavity changes is the inflection point — f″ = 0 there. Slide slowly and watch the tangent tip rotate accordingly.

f″ > 0 — concave up

Curve bowls upward like a cup. At a stationary point → minimum.

f″ < 0 — concave down

Curve caps over like an umbrella. At a stationary point → maximum.

f″ = 0 — possible inflection

Concavity may flip here. Alone it proves nothing — check the sign change around it.

Why businesses care about concavity

A profit curve that is concave down (f″ < 0) has a peak — push production to that peak and you stop. A cost curve that is concave up (f″ > 0) has a valley — that valley is your cheapest scale of production. Concavity is where “more is better” turns into “more is worse”.

Explore: Second Derivative Solver

Enter any polynomial and step through the full f″ workflow: differentiate → differentiate again → evaluate concavity at a slideable point → locate the inflection points. The curve is colour-coded by the sign of f″ as soon as step 2 unlocks.

Second Derivative Solver — f(x) = 3x⁴ − 2x³ + x² − 8x + 7

Work through steps 1–4: differentiate → differentiate again → evaluate f″ where you slide → locate the inflection points.

Slide the evaluation point — x = 1.00

How to read the graph

From step 2 the curve is painted by f″: green = concave up(f″ > 0), red = concave down(f″ < 0). The teal dashed line is the tangent at your slide point. Violet dots (step 4) mark the inflection points where the colour — and the bending — flips. Try f(x) = x³ − 3x: one inflection at x = 0, exactly like the explorer above.

The Max–Min Test

Combined with f′ = 0, the sign of f″ decides whether a stationary point is a peak or a valley.

Second order derivative test
Condition at x = a	Verdict	Shape
f′(a) = 0 and f″(a) < 0	Maximum	peak — bowl down
f′(a) = 0 and f″(a) > 0	Minimum	valley — bowl up
f′(a) = 0 and f″(a) = 0	Inconclusive	use the first derivative test

f″ = 0 is not an answer

If the second derivative vanishes at the stationary point the test fails — fall back to checking whether f′ changes sign around a (positive → negative = max, negative → positive = min).

Worked Examples

Solved Example

Problem

Obtain dy/dx and d²y/dx² for y = 3x⁴ − 2x³ + x² − 8x + 7. Also evaluate d²y/dx² at x = 1.

Solution

d²y/dx² at x = 1 is 26

Solved Example

Problem

If f(x) = 4x³ + 2x² + 7x + 9, for which value of x is f″(x) = 52?

Solution

x = 2

Solved Example

Problem

Find f″(x) if f(x) = ∜x (Section C practice).

Solution

f″(x) = −(3/16) x^(−7/4)

Key Takeaways

★ f″(x) = d/dx [f′(x)] — differentiate the derivative; never square the first derivative.
f″ > 0 → concave up (bowl ∪) · f″ < 0 → concave down (bowl ∩) · f″ = 0 → check for an inflection point.
★ Max–min test: f′ = 0 with f″ < 0 gives a maximum; f′ = 0 with f″ > 0 gives a minimum.
If f″ = 0 the test is inconclusive — fall back to the sign of f′ around the point.
So what? Cost minimisation, revenue maximisation and profit maximisation in the business topics are literally this test with rupee-shaped curves.

Second Order Derivative

Notes

What Is a Second Order Derivative?

Concavity: What f″ Tells You Geometrically

Explore: Second Derivative Solver

The Max–Min Test

Worked Examples

Key Takeaways

Related Questions

Study Simplify Special