Answer:
[0,1] and [-1,0]
Step-by-step explanation:
We have the function [tex]f(x)=-14x^4-7x^3-18x^2+17x+11[/tex] to determine consecutive values of x between which each real zero is located we have to replace the function with different values and then analyze the results.
We are going to replace the function with x=0,
[tex]f(0)=-14.(0)^4-7.(0)^3-18.(0)^2+17.(0)+11\\f(0)=11[/tex]
Now we are going to replace with x=1,
[tex]f(1)=-14.(1)^4-7.(1)^3-18.(1)^2+17.(1)+11\\f(1)=-14-7-18+17+11\\f(1)=-11[/tex]
With x=2,
[tex]f(2)=-14.(2)^4-7.(2)^3-18.(2)^2+17.(2)+11\\f(2)=-224-56-72+34+11\\f(2)=-307[/tex]
Now with x=-1,
[tex]f(-1)=-14.(-1)^4-7.(-1)^3-18.(-1)^2+17.(-1)+11\\f(-1)=-14+7-18-17+11\\f(-1)=-31[/tex]
With x=-2,
[tex]f(-2)=-14.(-2)^4-7.(-2)^3-18.(-2)^2+17.(-2)+11\\f(-2)=-224+56-72-34+11\\f(-2)=-195[/tex]
We have to analyze:
We have that f(2) and f(1) have the same signs both are negatives this means that there isn't a zero between the interval [1,2].
We have that f(1) and f(0) have opposite signs this means that there is a zero between the interval [0,1].
f(0) and f(-1) have opposite signs too, then there's also a zero in the interval [-1,0].
And finally, f(-1) and f(-2) have the same signs then there isn't a zero in the interval [-2,-1]
The graph of the function shows that the answer is correct:
