Suppose that, for x ≥ 0, the curve C has equation y = 3x − x√x.(a) Find the coordinates of the x-intercepts of C
Question
Suppose that, for x ≥ 0, the curve C has equation
y = 3x - x\sqrt{x}.
(a) Find the coordinates of the x-intercepts of C.
Solution
The x-intercepts of the curve C are the points where the curve intersects the x-axis. This happens when y = 0.
So, to find the x-intercepts, we set y = 0 in the equation of the curve and solve for x.
0 = 3x - x√x
We can factor out an x from each term on the right side of the equation:
0 = x(3 - √x)
Setting each factor equal to zero gives the solutions to the equation:
x = 0 or 3 - √x = 0
The first equation gives x = 0.
The second equation can be solved for x by first adding √x to both sides to get √x = 3, and then squaring both sides to get x = 9.
So, the x-intercepts of the curve C are (0, 0) and (9, 0).
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