Answer
(x,y)= (-1,-5/6) and (4,-5/3)
Work Step by Step
Rearrange the equation 8x-2y=1 become:
8x-1=2y
2y=8x-1
y=(8x/2)-(1/2)
y=4x-1/2.
From y=mx+c,
we know m=4.
Differentiate g(x)=(1/3)x^{3} - (3/2)x^{2} + 1
dg/dx= (1/3)(3)x^{3-1} - (3/2)(2)x^{2-1} +0
dg/dx= x^{2} - 6x
The tangent line is parallel to the line 8x-2y=1, thus slope of the tangent line = 4.
So, x^{2} - 6x = 4.
x^{2} - 6x -4 =0
Factorize the equation.
(x-4)(x+1)=0
so we get x-4=0 and x+1=0.
The first point is when x-4=0,
x=4.
To find y-coordinate, we substitute x with 4 in the g(x).
So, g(4)=(1/3)(4)^{3} - (3/2)(4)^{2} + 1
g(4)=-5/3.
The second point is when x+1=0,
x=-1,
Same as above, substitute x with -1 in g(x),
g(-1)=(1/3)(-1)^{3} - (3/2)(-1)^{2} + 1
g(-1)=-5/6
Therefore, (x,y)= (-1,-5/6) and (4,-5/3).
