Question

Let $q(t)$ denote the quadratic interpolating polynomial that goes through the data points $\left(t_0, y_0\right),\left(t_1, y_1\right),\left(t_2, y_2\right)$. (a) Under what conditions does $q(t)$ have a minimum? A maximum? (b) Show that the minimizing/maximizing value is at $t^{\star}=\frac{m_0 s_1-m_1 s_0}{s_1-s_0}$, where $s_0=\frac{y_1-y_0}{t_1-t_0}, s_1=\frac{y_2-y_1}{t_2-t_1}, m_0=\frac{t_0+t_1}{2}, m_1=\frac{t_1+t_2}{2}$. (c) What is $q\left(t^{\star}\right)$ ?

   Let $q(t)$ denote the quadratic interpolating polynomial that goes through the data points $\left(t_0, y_0\right),\left(t_1, y_1\right),\left(t_2, y_2\right)$. (a) Under what conditions does $q(t)$ have a minimum? A maximum? (b) Show that the minimizing/maximizing value is at $t^{\star}=\frac{m_0 s_1-m_1 s_0}{s_1-s_0}$, where $s_0=\frac{y_1-y_0}{t_1-t_0}, s_1=\frac{y_2-y_1}{t_2-t_1}, m_0=\frac{t_0+t_1}{2}, m_1=\frac{t_1+t_2}{2}$.
(c) What is $q\left(t^{\star}\right)$ ?

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 5, Problem 25

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Step 1

The coefficients $ a $, $ b $, and $ c $ are determined such that the polynomial passes through the given points $ (t_0, y_0) $, $ (t_1, y_1) $, and $ (t_2, y_2) $. Show more…

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Let $q(t)$ denote the quadratic interpolating polynomial that goes through the data points $\left(t_0, y_0\right),\left(t_1, y_1\right),\left(t_2, y_2\right)$. (a) Under what conditions does $q(t)$ have a minimum? A maximum? (b) Show that the minimizing/maximizing value is at $t^{\star}=\frac{m_0 s_1-m_1 s_0}{s_1-s_0}$, where $s_0=\frac{y_1-y_0}{t_1-t_0}, s_1=\frac{y_2-y_1}{t_2-t_1}, m_0=\frac{t_0+t_1}{2}, m_1=\frac{t_1+t_2}{2}$. (c) What is $q\left(t^{\star}\right)$ ?

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