= Solution
{c}
The <polynomial> $T_6$ is
$$
T_6(x)=32x^6-48x^4+18x^2-1.
$$
Subtract its monic version from $x^6$:
$$
\boxed{p(x)=x^6-\frac1{32}T_6(x)
=\frac32x^4-\frac9{16}x^2+\frac1{32}.}
$$
This has degree four and hence belongs to the usual space $P_5$. The error is $T_6/32$, with <norm> $1/32$ and alternating extrema at the seven points $\cos(j\pi/6)$, $j=0,\ldots,6$. Part (a), followed by <uniqueness>, proves
$$
\boxed{\|x^6-p\|_\infty=\frac1{32}}
$$
and that this is the unique <best approximation> of degree at most five.
If “degree five” is taken literally to exclude degree four, no <best approximation> exists in that nonclosed class: $p+\epsilon x^5$ approaches the error $1/32$ as $\epsilon\to0$, but <uniqueness> in $P_5$ excludes attainment by any degree-five <polynomial>. This is why the standard at-most-degree convention matters here.
Back to article page
