For what value of k does the parabola y= x2/3 −kx+3 intersect the x-axis in only one point?
y = ( x^2) / 3 - k x + 3 ????
solution to quadratic is if y = a x^2 + b x + c
y= 0 (hits x axis) when x = [-a +/- sqrt(b^2-4ac) ]/ 2a
only one solution if b^2 - 4 a c = 0
so
b^2-4 a c = 0
k^2 - 4 = 0
k^2 = 2 or -2
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To find the value of k for which the given parabola intersects the x-axis at only one point, we need to determine the discriminant of the quadratic equation.
The equation of the parabola is given by: y = x^(2/3) - kx + 3
When the parabola intersects the x-axis, the y-coordinate will be equal to zero. Therefore, we can set y = 0 and solve for x.
0 = x^(2/3) - kx + 3
To make further calculations easier, we can multiply the entire equation by x^(1/3):
0 = x - kx^(4/3) + 3x^(1/3)
Now, we have a cubic equation in terms of x. We need to determine the discriminant of this equation, which can be done using the concept of differentiation.
First, we differentiate the equation with respect to x:
0 = 1 - (4/3)kx^(1/3) + (1/3)x^(-2/3)
Now, we differentiate it again:
0 = -(4/9)kx^(-2/3) - (2/9)x^(-5/3)
Setting this equation equal to zero, we get:
0 = -(4/9)kx^(-2/3) - (2/9)x^(-5/3)
Since x cannot be zero (as it is in the denominator), we can divide the entire equation by x^(-5/3):
0 = -(4/9)k - (2/9)x^(-3/3)
0 = -(4/9)k - (2/9)x^(-1)
0 = -(4/9)k - (2/9)/x
We can rearrange this equation to get:
(4/9)k = -(2/9)/x
Simplifying further:
k = -1/2x
Now, we can see that for any value of x, there will be a corresponding value of k. Thus, there isn't a specific value of k for which the parabola intersects the x-axis at only one point.
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