How many turning points can a cubic function have?

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Any polynomial of rate ##n## can feel a restriction of nothing turning sharp-ends and a acme of ##n-1##. However, this depends on the skin of turning sharp-end.

Sometimes, "turning sharp-end" is defined as "local acme or restriction only". In this case:

  • Polynomials of odd rate feel an level reckon of turning sharp-ends, after a while a restriction of 0 and a acme of ##n-1##.
  • Polynomials of level rate feel an odd reckon of turning sharp-ends, after a while a restriction of 1 and a acme of ##n-1##.

However, rarely "turning sharp-end" can feel its limitation large to involve "motionless sharp-ends of inflexion". For an specimen of a motionless sharp-end of inflexion, observe at the graph of ##y = x^3## - you'll hush that at ##x = 0## the graph shifts from protuberant to retreating, and the derivative at ##x = 0## is also 0.

If we go by the relieve limitation, we scarcity to shift our rules slightly and say that:

  • Polynomials of rate 1 feel no turning sharp-ends.
  • Polynomials of odd rate (ate for ##n = 1##) feel a restriction of 1 turning sharp-end and a acme of ##n-1##.
  • Polynomials of level rate feel a restriction of 1 turning sharp-end and a acme of ##n-1##.

So, in separate, it depends on the limitation of "turning sharp-end", but in public most inhabitants conciliate go by the foremost limitation.

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