Concave: f =0 (cup shape). Opposite curvature directions.

Yes, f=-1/x^2 0. Strictly concave on its domain (0,inf). Classic example.

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Concave Function Checker

Check if f(x) <= 0 (concave) for common function types

Select Function

ax^2+bx+c ax^3+bx^2 e^(ax) ln(x)

f(x)=x^2+x+

Concave Function Test

f(x) <= 0 everywhere: CONCAVE

f(x) >= 0 everywhere: CONVEX

f changes sign: NEITHER

Linear functions: both

A concave function has a second derivative that is never positive. It curves downward like an upside-down cup. Concave functions are important in economics (diminishing returns, utility) and optimization (unique global maximum).

⚠ Concave = f(x) <= 0. f(x) < 0 means strictly concave. f(x) = 0 (linear) qualifies as both convex and concave.

What Is a Concave Function?

A function is concave if all chords lie below the graph. The second derivative test: f(x) <= 0 means concave. Examples: -x^2 (always concave), ln(x) (concave for x>0), sqrt(x) (concave). Concave functions have at most one global maximum.

Quadratic

f=2a. If a<0: always concave. If a>0: always convex. a=0: linear (both).

Exponential

e^(ax): f = a^2 * e^(ax) >= 0 always. e^(ax) is convex for any a. Never concave.

Logarithm

ln(x): f = -1/x^2 < 0 for x>0. Strictly concave on its domain. Classic concave example.

Cubic

f=6ax+2b. Changes sign at x=-b/(3a). Neither globally convex nor concave.

Teaching Example: f(x)=-x^2. f=-2<0. f(x) <= 0 everywhere -> CONCAVE (sheds water, has global max at 0).

Applications

Economics Optimization Utility Theory Physics Statistics

Frequently Asked Questions

What is concave?▼

f(x) <= 0. Curves downward (cap). -x^2, ln(x) are concave. Chords lie below the graph.

Concave vs convex?▼

Concave: f<=0 (cap shape). Convex: f>=0 (cup shape). Opposite curvature directions.

Is ln(x) concave?▼

Yes, f=-1/x^2<0 for x>0. Strictly concave on its domain (0,inf). Classic example.

Is x^3 concave?▼

No, f=6x changes sign. x<0: concave (f<0). x>0: convex (f>0). Neither globally.

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