Exercise 1.3.4

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ewcommand{\ssi} {si et seulement si }

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Finally, consider the case $p<0$. In this case $f'(y) = 3y^2+p$ has roots $\alpha = \sqrt{-p/3}$ and $\beta = - \sqrt{-p/3}$, which are real and distinct.
\begin{enumerate}
\item[(a)] Show that the graph of $f(y)$ has a local minimum at $\alpha$ and a local maximum at $\beta$. Thus $f(\alpha)$ is a local minimum value and $f(\beta)$ is a local maximum value. Also show that $f(\alpha)<f(\beta)$.
\item[(b)] Explain why $f(y)$ has three real roots if $f(\alpha)$ and $f(\beta)$ have opposite signs and has one real root if they have the same sign. Illustrate your answer with a drawing of the three cases that can occur.
\item[(c)] Conclude that $f(y)$ has three real roots if and only if $f(\alpha)f(\beta)<0$.
\item[(d)] Finally, use part (c) of Exercise 1 to show that the roots are all real if and only if $\Delta >0$.

\end{enumerate}

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow} \def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers \def\dcro{\mbox{]\hspace{-.15em}]}} ewcommand{\be} {\begin{enumerate}} ewcommand{\ee} {\end{enumerate}} ewcommand{\deb}{\begin{eqnarray*}} ewcommand{\fin}{\end{eqnarray*}} ewcommand{\ssi} {si et seulement si } ewcommand{\D}{\mathrm{d}} ewcommand{\Q}{\mathbb{Q}} ewcommand{\Z}{\mathbb{Z}} ewcommand{\N}{\mathbb{N}} ewcommand{\R}{\mathbb{R}} ewcommand{\C}{\mathbb{C}} ewcommand{\F}{\mathbb{F}} ewcommand{\U}{\mathbb{U}} ewcommand{ e}{\,\mathrm{Re}\,} ewcommand{\im}{\,\mathrm{Im}\,} ewcommand{\ord}{\mathrm{ord}} ewcommand{\Gal}{\mathrm{Gal}} ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}} \begin{proof} Case $p<0$. \begin{enumerate} \item[(a)] $f'(y) = 3y^2+p < 0 \iff y^2<-p/3 \iff \beta < f'(y) < \alpha\ (\alpha = \sqrt{-p/3} = -\beta)$. $f$ is strictly increasing on $]-\infty,\beta]$ and on $[\alpha, +\infty[$, strictly decreasing on $[\beta,\alpha]$. $\beta$ is a local maximum, and $\alpha$ a local minimum. As $f$ is decreasing on $[\beta,\alpha]$, $f(\beta) > f(\alpha)$. \item[(b)] As $f$ is continuous, and $\lim\limits_{y o +\infty} f(y) = +\infty$, $\lim\limits_{y o -\infty} f(y) = -\infty$ and $f$ strictly monotonous on each interval $]-\infty,\beta],[\beta,\alpha],[\alpha, +\infty[$, the intermediate value theorem gives three roots if $f(\alpha)f(\beta)<0$, and a unique root if $f(\alpha) f(\beta)>0$. If $f(\alpha)f(\beta) = 0$, then $\Delta = -27 f(\alpha)f(\beta) = 0$ (Ex.1 c), which we can exclude by hypothesis. \item[(c)] As these cases are mutually exclusive, (b) proves that $f$ has 3 real roots iff $f(\alpha)f(\beta)<0$. \item[(d)] By Ex. 1.3.2 and 1.3.3, we know that $p\geq 0$ imply that $f$ has a unique real root. If $f$ has 3 distinct real roots, then $\Delta eq 0$ (Ex.1.2.4), $p<0$, and (c) shows that $f(\alpha)f(\beta) <0$, thus $\Delta = -27 f(\alpha)f(\beta) >0$. Conversely, if $\Delta>0$, then $p<0$ and $f(\alpha)f(\beta)<0$. By (c), we know that $f$ has 3 real roots. Conclusion : $f$ has three distinct real roots iff $\Delta >0$. \end{enumerate} \end{proof}

Exercise 1.3.4

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Exercise 1.3.4

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