Suppose that, for [tex]\( x \geq 0 \)[/tex], the curve [tex]\( C \)[/tex] has the equation [tex]\( y = 3x - x \sqrt{x} \)[/tex].

(a) Find the coordinates of the [tex]\( x \)[/tex]-intercepts of [tex]\( C \)[/tex].



Answer :

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].