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Extrema Calculator

Find minima and maxima of quadratic and cubic functions

Select Function Type

Quadratic ax^2+bx+c Cubic ax^3+bx^2

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

f(x)=x^2 +x +

Extrema Rules

Quadratic vertex: x = -b/(2a), y = f(-b/(2a))

a > 0: vertex is minimum (opens up)

a < 0: vertex is maximum (opens down)

Cubic: set derivative f(x)=0 for critical points

Function extrema are the highest and lowest points. Quadratic functions have exactly one global extremum at the vertex. Cubic functions can have two local extrema where the derivative equals zero.

⚠ The vertex formula x=-b/(2a) only works for quadratic functions. For other functions, use derivative-based methods.

What Are Extrema?

Extrema are the extreme values of a function. For quadratics, the vertex formula directly gives the extremum. For cubics, we find critical points by taking the derivative and solving f(x)=0. The second derivative test classifies each as min or max.

Quadratic Vertex

x=-b/(2a), y=f(x). a>0 min, a<0 max. Axis of symmetry through vertex. Global extremum.

Cubic Critical Points

Derivative f(x)=3ax^2+2bx=0 -> x=0 or x=-2b/(3a). Use f(x) to classify.

Second Derivative Test

If f(x)>0, critical point is local min. If f(x)<0, local max. If f(x)=0, test inconclusive.

Global vs Local

Quadratic vertex is always global. Cubic local extrema may not be global as cubics go to +/- inf.

Teaching Example: f(x)=x^2-4x+3. Vertex: x=-(-4)/(2*1)=2. f(2)=4-8+3=-1. Since a=1>0, vertex (2,-1) is minimum.

Applications

Optimization Physics Economics Engineering Graphing

Frequently Asked Questions

What are extrema?▼

Minima and maxima of a function. For quadratics: vertex = x=-b/(2a). For cubics: critical points from derivative.

Vertex formula?▼

x=-b/(2a). Vertex is minimum if a>0, maximum if a<0. The turning point of the parabola.

How many extrema?▼

Quadratic: exactly 1 (vertex). Cubic: up to 2 local extrema (where derivative=0). Higher degrees may have more.

Local vs global?▼

Local: highest/lowest in neighborhood. Global: absolute highest/lowest overall. Quadratic vertex is always global.

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