1

Problem

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

2

Step

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

3

Step

Given the cubic function

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

4

Step

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\$

5

Step

To find the remaining roots, then solve the quadratic equation

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

6

Formula

The quadratic formula is \$((- b) ± \sqrt(b^2 - 4ac))/(2a)\$.

7

Hint

Where a = 1, b = − 6, and c = 9.

8

Step

Now plug the values in the quadratic formula then after simplification

x = \$(−(−6)) ± \sqrt((−6)^2−4(1)(9))/2(1)\$

x = \$(6) ± \sqrt(36 − 36)/2\$

x = 3

9

Solution

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

10

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

11

Choice-A

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

Wrong x = (0, 1)

12

Choice-B

This option is not correct, because(0, 2)is not an intercept since substituting x=2 into the function does not result in 0

Wrong x = (0, 2)

13

Choice-C

This option is correct, because x-intercepts on the list correspond accurately with the roots of the cubic function

Correct x = (0, 3)

14

Choice-D

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

Wrong x = (0, 4)

15

Answer

Option

C

16

Sumup

Please summarize choices