Answer: The required zeroes of the given function are 0, 3 and 4.
Step-by-step explanation: We are given to find the zeroes of the following function :
[tex]f(x)=3x^3-21x^2+36x~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We know that the zeroes of the function y = f(x) are given by f(x) = 0.
Therefore, from equation (i), we get
[tex]f(x)=0\\\\\Rightarrow 3x^3-21x^2+36x=0\\\\\Rightrarow 3x(x^2-7x+12)=0\\\\\Rightarrow 3x=0,~~x^2-7x+12=0\\\\\Rightarrow x=0,~~\Rightarrow x^2-3x-4x+12=0\\\\\Rightarrow x=0,~~\Rightarrow x(x-3)-4(x-3)=0\\\\\Rightarrow x=0,~~\Rightarrow (x-3)(x-4)=0\\\\\Rightarrow x=0,~~x-3=0,~~x-4=0\\\\\Rightarrow x=0,3,4.[/tex]
Thus, the required zeroes of the given function are 0, 3 and 4.
