Question
Show that every polynomial function of degree 3 $f(x)=a x^{3}+b x^{2}+c x+d$ has exactly one inflection point.
Step 1
The derivative of a function gives us the slope of the tangent line at any point on the function. The first derivative of the function is given by: \[f'(x) = 3ax^{2} + 2bx + c\] Show more…
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Show that the general quartic (fourth-degree) polynomial $f(x)=x^{4}+a x^{3}+b x^{2}+c x+d$ has either zero or two inflection points, and the latter case occurs provided that $b<3 a^{2} / 8.$
Show that the general quartic (fourth-degree) polynomial $f(x)=x^{4}+a x^{3}+b x^{2}+c x+d$ has either zero or two inflection points, and the latter case occurs provided that $b<3 a^{2} / 8$.
What Derivatives Tell Us
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