Answer
Please see proof in "step by step"
Work Step by Step
The Intermediate Value Theorem for Polynomial Functions, p.355:
Let $f$ be a polynomial function with real coefficients.
If $f(a)$ and $f(b)$ have opposite signs,
then there is at least one value of $c$ between $a$ and $b$ for which $f(c)=0$.
Equivalently, the equation $f(x)=0$ has at least one real root between $a$ and $b$.
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(c is a real number in the above theorem)
$f(1)=1^{3}-2(1)-1=-2 < 0$
$f(2)=2^{3}-2(2)-1=5 > 0,$
If we substitute a=1, b=2 in the Theorem, we conclude that
since f(1) and f(2) have different signs,
there is at least one real root of $f(x)=0$ between 1 and 2.
that is, f has at least one real zero between 1 and 2.
