Given:
The function is
[tex]f(x)=x^3-3x^2+5x-15[/tex]
To find:
The root of the function from the given possible roots.
Solution:
We have,
[tex]f(x)=x^3-3x^2+5x-15[/tex]
At x=-3,
[tex]f(-3)=(-3)^3-3(-3)^2+5(-3)-15[/tex]
[tex]f(-3)=-27-27-15-15[/tex]
[tex]f(-3)=-84\neq 0[/tex]
At x=-1,
[tex]f(-1)=(-1)^3-3(-1)^2+5(-1)-15[/tex]
[tex]f(-1)=-1-3-5-15[/tex]
[tex]f(-1)=-24\neq 0[/tex]
At x=1,
[tex]f(1)=(1)^3-3(1)^2+5(1)-15[/tex]
[tex]f(1)=1-3+5-15[/tex]
[tex]f(1)=-12\neq 0[/tex]
At x=3,
[tex]f(3)=(3)^3-3(3)^2+5(3)-15[/tex]
[tex]f(3)=27-27+15-15[/tex]
[tex]f(3)=0[/tex]
Since, the value of given function is 0 at only x=3, therefore 3 is a root of given function.
Hence, the correct option is D.
