determine value of k when g(x)= 4x+k does not intersect the parabola f(x)=-3x2-x+4
The function g(x) is a straight line and the function f(x) is a downward opening parabola.
The marginal condition is that the line is tangent to the parabola. For this condition there is just one point of intersection. We can find the condition for just one point of intersection by equating the two functions. After setting f = g and reorganizing, we find
3 x2 +5 x - 4 + k = 0
We want there to be just one solution to this. Since this is a quadratic form, we compute the discriminant D.
D = 52 - 4 (3 ) (-4 +k). There will be just one solution when D = 0. This gives a condition for k
k = 73/12. For k > 73/12 the line will be always above the downward opening parabola.
For k < 73/12 the line will intersect the parabola twice.
Thus the answer we are looking for is k > 73/12