To find the [tex]\(x\)[/tex]-intercepts of the curve [tex]\(C\)[/tex] defined by the equation [tex]\(y = 3x - x\sqrt{x}\)[/tex], we need to determine the values of [tex]\(x\)[/tex] where [tex]\(y = 0\)[/tex].
1. Starting with the equation:
[tex]\[
y = 3x - x\sqrt{x}
\][/tex]
2. Set [tex]\(y\)[/tex] to zero to find the [tex]\(x\)[/tex]-intercepts:
[tex]\[
0 = 3x - x\sqrt{x}
\][/tex]
3. Factor out [tex]\(x\)[/tex] from the right-hand side:
[tex]\[
0 = x (3 - \sqrt{x})
\][/tex]
This equation implies that the product of [tex]\(x\)[/tex] and [tex]\((3 - \sqrt{x})\)[/tex] is zero. Therefore, we set each factor to zero separately to solve for [tex]\(x\)[/tex].
4. Solve the first factor:
[tex]\[
x = 0
\][/tex]
5. Solve the second factor:
[tex]\[
3 - \sqrt{x} = 0
\][/tex]
6. Isolating [tex]\(\sqrt{x}\)[/tex]:
[tex]\[
\sqrt{x} = 3
\][/tex]
7. Squaring both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 9
\][/tex]
Since [tex]\(x \geq 0\)[/tex], the [tex]\(x\)[/tex]-intercepts are [tex]\(x = 0\)[/tex] and [tex]\(x = 9\)[/tex].
8. Now write the coordinates of the [tex]\(x\)[/tex]-intercepts:
[tex]\[
(0, 0) \quad \text{and} \quad (9, 0)
\][/tex]
Thus, the coordinates of the [tex]\(x\)[/tex]-intercepts of the curve are [tex]\((0, 0)\)[/tex] and [tex]\((9, 0)\)[/tex].
