To find the positive points at which local minimums of
occur, let's find the derivative of
first.
Since
can be written as a composite of two functions,
, where
, we can apply the chain rule to find its derivative. In this process, we need the derivative of
.
By the Fundamental theorem of Calculus, we have
![{\displaystyle F'(x)={\frac {d}{dx}}\left(\int _{0}^{x}(t^{4}+1)\sin {t}dt\right)=(x^{4}+1)\sin x.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/a5c60e55e44068561bc96423ef838c4ed006d6d5)
Then, the derivative of
is
![{\displaystyle g'(x)=(F(h(x))'=F'(h(x))h'(x)=F'(x^{2})(x^{2})'=2x(x^{8}+1)\sin(x^{2}).}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/8a6177e07a6275a31d4027c1bedbd6b8e5ba00c3)
Since
for any
on the real line and we consider only positive
, the sign of
is determined by
.
Note that
at
(i.e.,
). Also,
![{\displaystyle {\begin{aligned}&\sin(x^{2})>0,\quad {\text{on }}(0,{\sqrt {\pi }}),({\sqrt {2\pi }},{\sqrt {3\pi }}),\cdots \\&\sin(x^{2})<0,\quad {\text{on }}({\sqrt {\pi }},{\sqrt {2\pi }}),({\sqrt {3\pi }},{\sqrt {4\pi }}),\cdots .\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/d2cd0d8e509aa0a5c29091f8aa5ace38970d3be8)
Since the sign of
is changed from minus to plus at
for the first time among positive
, so is the sign of
. Therefore, the smallest number
at which
has a local minimum is
.
Answer: