Consider the cubic function
\$f(x) = x^3 - 6x^2 + 9x\$. Let’s find the x-intercepts of this function.

To find the x-intercepts of a function, we need to determine the values of x for which the function f(x) equals zero.

Given the cubic function

\$ f(x) = x^3 - 6x^2 + 9x \$

Let’s set f(x) equal to zero, then factor out the common term x from each term

\$ x^3 - 6x^2 + 9x = 0 \$

\$ x(x^2 - 6x + 9) = 0 \$

\$ x = 0,x^2 - 6x + 9 = 0 \$

To find the remaining roots, then solve the quadratic equation

\$ x^2 - 6x + 9 = 0 \$

Quadratic formula

\$ x = ( (-b) pm \sqrt(b^2 - 4ac))/(2a) \$

where a = 1, b = − 6, and c = 9

Now plug the values in the quadratic formula then after simplification

\$ x = ( (-(-6)) pm \sqrt((-6)^2 - 4(1)(9)))/(2(1)) \$

\$ x = ( 6 pm \sqrt(36 - 36))/(2) \$

\$ x = 3 \$

So, the x-intercepts of the function \$f(x) = x^3 - 6x^2 + 9x\$ are \$ x = 0 and x = 3 \$.

This is incorrect as the function does not intersect the x-axis at x = 1, since substituting 1 into the function does not yield 0

x = 0 , 1

This is not an intercept since substituting x=2 into the function does not result in 0

x = 0 , 2

The x-intercepts on the list correspond accurately with the roots of the cubic function

x = 0 , 3

This choice is Incorrect. Substituting x = 4 into the function does not equal 0, indicating it’s not an x-intercept

x = 0 , 4

Option

C

Can you summarize what you’ve understood in the above steps?