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 $b<3 a^{2} / 8.$
Added by Dale M.
Step 1
The second derivative is given by: $$f''(x) = \frac{d^2}{dx^2}(x^4 + ax^3 + bx^2 + cx + d)$$ Differentiating each term separately, we get: $$f''(x) = \frac{d^2}{dx^2}(x^4) + \frac{d^2}{dx^2}(ax^3) + \frac{d^2}{dx^2}(bx^2) + \frac{d^2}{dx^2}(cx) + Show more…
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