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Relative Extrema Finder

Find local minima and maxima using derivative tests

Select Function Type

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

f(x)=x^2 +x +

Relative Extrema Tests

First derivative test: sign change of f

Second derivative test: f>0 min, f<0 max

Quadratic: exactly one (vertex)

Cubic: up to two (at f=0 points)

Relative extrema are local peaks and valleys. Use critical points where f=0. The first derivative test checks the sign of f on either side. The second derivative test checks concavity at the critical point.

⚠ Not every critical point is an extremum. Saddle points (f=0 but no sign change) are not local extrema.

What Are Relative Extrema?

Relative (local) extrema are function values that are extreme within a small region. A local max is higher than all nearby points. A local min is lower. For quadratics, the vertex is always a relative extremum. Cubics can have two local extrema.

Quadratic

One critical point at x=-b/(2a). f=2a constant. If a>0 -> local min. a<0 -> local max. Always an extremum.

Cubic

f=3ax^2+2bx=0 gives up to 2 critical points. Use second derivative test to classify each as min or max.

First Derivative Test

f from + to - : local max. f from - to + : local min. f no change: saddle point (not extremum).

Second Derivative Test

At f=0: f>0 means concave up -> min. f<0 means concave down -> max. f=0: test inconclusive.

Teaching Example: f(x)=x^2-4x+3. f=2x-4=0 at x=2. f=2>0. Second derivative test: f>0 -> local MIN at (2,-1).

Applications

Optimization Curve Sketching Physics Economics Machine Learning

Frequently Asked Questions

What is relative extrema?▼

Local max/min. Higher/lower than nearby points. Found at critical points where f=0 and sign changes.

First derivative test?▼

Check f sign on both sides of critical point. + to - = max. - to + = min. No change = saddle.

Second derivative test?▼

At f=0: if f(>0 min, <0 max, =0 inconclusive. Faster than first derivative test when it works.

All critical points extrema?▼

No. Some are saddle points where f does not change sign (e.g., f(x)=x^3 at x=0). Always verify.

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