# Negative value of $\sqrt[3]{20}$

Given $f(x)=\sqrt[3] x$, find an approximation of $\sqrt[3]{20}$ using Lagrange interpolation method.
$x_0=0$, $x_1=1$, $x_2=8$, $x_3=27$ and $x_4=64$
$f(x_0)=0$, $f(x_1)=1$, $f(x_2)=2$, $f(x_3)=3$ and $f(x_4)=4$

What I've done is, calculating $p(20)=-1.3139$ which is obviously wrong. Why is that?

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It's basically a cautionary tale for interpolating polynomials. They don't necessarily have to behave qualitatively like the functions they're supposed to be approximating. –  Ｊ. Ｍ. Feb 14 '12 at 12:42

That's a rather common problem with interpolating polynomials. For starters, the function you are approximating has a vertical tangent. Since polynomials can't have vertical tangents, any attempt to approximate the cube root with an interpolating polynomial will be troublesome. The other thing is that even polynomials of modest degree tend to be wiggly in between widely spaced interpolation points (except in some special cases), so the inaccuracy you observed is par for the course.

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