Answer
$f(1.4)$ and $f(1.5)$ attain opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[1.4, 1.5]$
Work Step by Step
We are given that $f(x)=x^5-x^4+7x^3-7x^2-18x+18$
The Intermediate Value Theorem states that when a function is continuous on an interval $[p,q]$ and takes on values $f(p)$ and $f(q)$ at the endpoints, then the function takes on all values between $f(p)$ and $f(q)$ at some point of the interval.
We will evaluate the function at the endpoints $[1.4, 1.5]$.
$f(1.4)=(1.4)^5-(1.4)^4+7(1.4)^3-7(1.4)^2-18(1.4)+18=-0.17536$
$f(1.5)=(1.5)^5-(1.5)^4+7(1.5)^3-7(1.5)^2-18(1.5)+18=1.40625$
This shows that $f(1.4)$ and $f(1.5)$ attain opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[1.4, 1.5]$.
