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(a) Given a function f that satisfies f(-1) = 3, f(0) = -4, f(1) = 5 and f(2)=-6. Find the unique Lagrange interpolating polynomial of degree 3 that agrees with all given function values. Show your steps. (b) Given a closed and bounded interval [c, d], and suppose this interval consists of (n + 1) regularly spaced nodes (including the two endpoints, c and d). Denote the nodes as {X1, X2, X3,..., Xn+1}. n+1 Prove that for all x € [c, d], [(x − x₁)(x − x₂) ... (x − Xn+1)| ≤n! ⋅ (d = c) (c) Let f(x) = sinx, and 4 nodes are given (including the two endpoints), at x = 0,7, and An interpolating polynomial of degree 3, P3 (x) is used to approximate f(x) on [0,1]. Show that when P3(x) is used to "approximate" f(x) at x = 2, the maximum possible error is less than 0.003. Show your steps. 12

(a) Given a function $f$ that satisfies $f(−1)=3,f(0)=−4,f(1)=5$ and $f(2)=−6$. Find the unique Lagrange interpolating polynomial of degree 3 that agrees with all given function values. Show your steps. (b) Given a closed and bounded interval $[c,d]$, and suppose this interval consists of $(n+1)$ regularly spaced nodes (including the two endpoints, $c$ and $d$ ). Denote the nodes as ${x_{1},x_{2},x_{3},…,x_{n+1}}$. Prove that for all $x∈[c,d],∣(x−x_{1})(x−x_{2})…(x−x_{n+1})∣≤n!⋅(nd−c )_{n+1}$ (c) Let $f(x)=sinx$, and 4 nodes are given (including the two endpoints), at $x=0,6π ,3π $ and $2π $. An interpolating polynomial of degree $3,P_{3}(x)$ is used to approximate $f(x)$ on $[0,2π ]$. Show that when $P_{3}(x)$ is used to "approximate" $f(x)$ at $x=12π $, the maximum possible error is less than 0.003 . Show your steps.

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