Studying functions

Concavity and inflection

The second derivative describes how the first derivative changes. Use its sign to distinguish upward and downward bending.

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1. Concavity changes sign

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Example graph

f=x^3,\quad f\prime=3x^2,\quad f\prime\prime=6x

For x³, f″=6x is negative left of zero and positive right of zero. The origin is an inflection point because the concavity changes.

f′(a): 3
f″(a): 6

The solid curve is f; the dashed line is its tangent at a. Read f(a) as height and f′(a) as slope. Questions use the input stated in their text.

2. Zero without inflection

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Example graph

f=x^4,\quad f\prime\prime=12x^2

For x⁴, f″=12x² is nonnegative and positive away from zero. It does not change sign at zero, so the curve stays concave up and has no inflection there.

f′(a): 4
f″(a): 12

The solid curve is f; the dashed line is its tangent at a. Read f(a) as height and f′(a) as slope. Questions use the input stated in their text.

3. Increasing but concave down

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Example graph

f=\ln x,\quad f\prime=\frac1x,\quad f\prime\prime=-\frac1{x^2}

For ln(x), f′=1/x>0 and f″=−1/x²<0 on x>0. The graph rises while its tangent slope decreases. Increasing and concave up are different properties.

f′(a): 1
f″(a): -1

The solid curve is f; the dashed line is its tangent at a. Read f(a) as height and f′(a) as slope. Questions use the input stated in their text.