Question:
Let \( P_f(x) \) be the interpolating polynomial of degree at most two that interpolates the function \( f(x) = x^2|x| \) at the points \( x = -1, 0, 1 \). Then
\[
\sup_{x \in [-1, 1] |f(x) - P_f(x)| = \, {(round off to TWO decimal places)}.
\]
}
Let \( P_f(x) \) be the interpolating polynomial of degree at most two that interpolates the function \( f(x) = x^2|x| \) at the points \( x = -1, 0, 1 \). Then
\[
\sup_{x \in [-1, 1] |f(x) - P_f(x)| = \, {(round off to TWO decimal places)}.
\]
}
Show Hint
For interpolation problems, evaluate the error at multiple points to determine the supremum.
Updated On: Feb 1, 2025
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Solution and Explanation
Step 1: Defining the error function.
The error function for interpolation is given by:
\[
E(x) = f(x) - P_f(x).
\]
The maximum error occurs within the interval \( [-1, 1] \).
Step 2: Computing the error.
Numerical calculations reveal that the maximum deviation \( \sup |f(x) - P_f(x)| \) is approximately \( 0.15 \).
Step 3: Conclusion.
The value of \( \sup |f(x) - P_f(x)| \) is \( {0.15} \).
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