Answer
True.
Work Step by Step
See sec.1-8, Intermediate Value Theorem.
If f is continuous on [a,b], N amy number between f(a) and f(b), f(a) $\neq$ f(b).Then, there exists a c in [a,b] such that f(c)=N.
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$f(x)=x^{10}-10x^{2}+5$
is a polynomial, so it is continuous everywhere.
$f(0)=5 > 0$
$f(2)=1024-400+5 > 0 ,$
which does not help, as we can not apply the Intermediate Value Theorem for [0,2].
But,
$f(1)=1-10+5 < 0$,
so, there will be a c in $[0,1]$ such that $f(c)=0.$
$c\neq 0, \ \ c\neq 1$ and $\ \ c\in(0,2)$,
So,
the statement is true.
