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Strictly Concave Analyzer

Check if f(x) < 0 (strictly concave) for common functions

Select Function

ax^2+bx+c ln(x) Linear ax+b -e^(ax)

f(x)=x^2+x+

Strict Concavity

f(x) < 0: STRICTLY CONCAVE

f(x) <= 0: concave (not strictly)

f(x) > 0: convex (not concave)

Unique global maximum guaranteed

Strict concavity requires f(x) < 0 at every point. This ensures a unique global maximum. Linear functions (f=0) are concave but not strictly concave. Strict concavity is a stronger condition used in optimization and economic theory.

⚠ Strictly concave requires f(x) < 0 everywhere. Linear functions (f=0) are concave but NOT strictly concave.

What Is Strict Concavity?

Strict concavity is stronger than concavity. It requires f(x) < 0 at every point. Strictly concave functions have a unique global maximum with no flat sections. They are the opposite of strictly convex functions and are important in utility theory and risk analysis.

Strictly Concave

-x^2 (f=-2<0), ln(x) (f=-1/x^2<0), -e^x (f=-e^x<0). Always curves down.

Concave but Not Strict

Linear functions: f=0. Concave (f<=0) but not strictly (f never < 0).

Not Concave

x^2 (f>0: convex). x^3 (f changes sign). e^x (f>0: convex).

Unique Maximum

Strict concavity guarantees a unique global maximum. Ensures well-behaved optimization.

Teaching Example: f(x)=-x^2. f=-2<0 always -> STRICTLY CONCAVE. Unique global max at (0,0). f(x)=3x+1. f=0 -> concave but NOT strictly.

Applications

Economics Utility Optimization Risk Analysis Statistics

Frequently Asked Questions

Strictly concave meaning?▼

f(x) < 0 everywhere. Unique global max. -x^2, ln(x) are strictly concave. Linear is not.

Concave vs strictly concave?▼

Concave: f<=0. Strict: f<0. Linear is concave but not strict. -x^2 is strictly concave.

Is ln(x) strictly concave?▼

Yes. f=-1/x^2 < 0 for all x>0. ln(x) is strictly concave on its domain.

Is -e^x strictly concave?▼

Yes. f=-e^x < 0 for all x. -e^x is strictly concave (negated exponential curves down).

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