Find local minima and maxima using derivative tests
Select Function Type
f(x)=x^2 +x +
f(x)=x^3 +x^2
Result
Derivation
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).
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