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Concavity Analyzer

Find intervals of concave up and concave down using second derivative analysis

Select Function Type

Quadratic ax^2+bx+c Cubic ax^3+bx^2+cx+d Quartic ax^4+bx^3

f(x)=ax^2+bx+c

f(x)=x^2+x+

Concavity Rules

f(x) > 0: concave up (U shape)

f(x) < 0: concave down (cap shape)

Quadratic: f=2a (constant, sign=sign of a)

Cubic: f=6ax+2b (linear, one zero)

Concavity describes the curvature of a graph. The second derivative test uses the sign of f(x) to determine whether a function bends upward or downward. Concavity changes at inflection points.

⚠ Concave up = f>0. Concave down = f<0. Remember: cup (positive) vs cap (negative) for the shape.

What Is Concavity?

Concavity measures how a graph bends. The second derivative f(x) is the rate of change of the slope. Positive f means the slope is increasing (concave up). Negative f means the slope is decreasing (concave down).

Quadratic

f=2a constant. If a>0 always concave up globally. If a<0 always concave down. No changes.

Cubic

f=6ax+2b linear. One zero divides domain: concave down on one side, concave up on the other.

Second Derivative Test

At critical point f=0: if f>0 -> local min (cup). If f<0 -> local max (cap). Useful for optimization.

Graph Interpretation

f>0: graph holds water (cup). f<0: graph sheds water (cap). Inflection at transition between them.

Teaching Example: f(x)=x^3-3x^2. f=6x-6. x<1: f<0 (concave down). x>1: f>0 (concave up). At x=1: inflection point.

Applications

Calculus Curve Sketching Optimization Physics Economics

Frequently Asked Questions

What is concavity?▼

Curvature direction. f>0: concave up (cup). f<0: concave down (cap). f measures slope change rate.

Quadratic concavity?▼

Constant: f=2a. a>0 always up, a<0 always down. No concavity changes for quadratics.

Cubic concavity?▼

f linear: 6ax+2b. Changes at x=-b/(3a). One concave up interval, one concave down interval.

Concavity and extrema?▼

At f=0: f>0 means min (cup). f<0 means max (cap). This is the second derivative test.

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