Answer
$f(1.7)$ and $f(1.8)$ attain opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[1.7, 1.8]$.
Work Step by Step
We are given that $f(x)=x^5-3x^4-2x^3+6x^2+x+2$
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 $[1.7, 1.8]$.
$f(1.7)=(1.7)^5-3(1.7)^4-2(1.7)^3+6(1.7)^2+1.7+2=0.35627$
$f(1.8)=(1.8)^5-3(1.8)^4-2(1.8)^3+6(1.8)^2+1.8+2=-1.0212$
This shows that $f(1.7)$ and $f(1.8)$ attain opposite signs. So, as per the Intermediate Value Theorem, there must be a real zero in the interval $[1.7, 1.8]$.
