Second Derivative & Inflection

Reading the Direction of Bending

Point position x1

👀 See It

①If f″>0 the curve is concave up (∪)

②If f″<0 it is concave down (∩)

③The point where concavity flips is the inflection point

④As you move the point, the spot where the color changes is the inflection point

Defining Concavity

Concavity test

f″(x) > 0 ⇒ concave up, f″(x) < 0 ⇒ concave down

The sign of the second derivative decides which way the curve bends

Condition for an inflection point

f″(a)=0 and f″ changes sign across x=a ⇒ (a, f(a)) is an inflection point

f″=0 is only necessary — you must also confirm the sign change

🪞 The Sign of f″ Sets the Shape

①Where f″(x)>0 the graph is concave up

②Where f″(x)<0 it is concave down

③An inflection point is where f″=0 AND the sign changes — f″=0 alone is not enough

Second-Derivative Test for Extrema

Second-derivative test

f'(a)=0, f″(a)>0 ⇒ local min / f″(a)<0 ⇒ local max

At a critical point, the sign of f″ quickly tells local max from min

Compute It Directly

Example 1

Find the inflection point of f(x)=x³−3x²+1.

Find the second derivative.

f'(x)=3x²−6x, f″(x)=6x−6

Solve f″(x)=0, confirm the sign change, then find the y-coordinate.

f″(x)=0 ⇒ x=1, f(1)=1−3+1=−1

▸ (1, −1)

Across x=1, f″ changes from − to +, so it is indeed an inflection point.

Example 2

Use the second-derivative test to find the extrema of f(x)=x³−3x.

From f'(x)=3x²−3=0 we get x=±1, and f″(x)=6x.

f″(1)=6>0 gives a local min; f″(−1)=−6<0 gives a local max.

local min f(1)=−2, local max f(−1)=2

▸ local max 2, local min −2

Without a sign chart for f′, a single sign of f″ separates max from min.

Wrap-up

Key result

f″>0 concave up, f″<0 concave down, sign-change point = inflection

Sketch a graph by reading f′ (monotonicity) and f″ (concavity·inflection) together

2021 KICE mock exam Math (Calculus) type, adapted

What is the inflection point of f(x)=x³−6x²+9x+1?

①(1, 5)

②(2, 3)

③(2, 5)

④(3, 1)

⑤No inflection point

▸ ② (2, 3)

f'(x)=3x²−12x+9, f″(x)=6x−12.

f″(x)=0 ⇒ x=2, f(2)=8−24+18+1=3.

inflection point (2, 3)

🎯 Exam Points

①f″>0 concave up, f″<0 concave down

②An inflection needs f″=0 AND a sign change (both)

③For extrema, set f'=0 then judge by the sign of f″

④Sketch via a sign chart of f′ and f″

⑤Always find the y-value of the inflection point too

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