To determine the values of k for which the function f(x) = kx^2 - 4x + k has no zeros, we can use the discriminant of the quadratic equation.
The quadratic equation is given by ax^2 + bx + c = 0, where in our case, a = k, b = -4, and c = k.
The discriminant (D) of a quadratic equation is calculated as D = b^2 - 4ac.
For the function f(x) = kx^2 - 4x + k to have no zeros, the discriminant D must be negative since the equation does not intersect the x-axis.
Substituting the values into the discriminant formula, we get D = (-4)^2 - 4(k)(k).
Simplifying, we have D = 16 - 4k^2.
To have no zeros, D < 0, so 16 - 4k^2 < 0.
Now, let's solve for k:
16 - 4k^2 < 0
Dividing both sides by 4, we get:
4 - k^2 < 0
Rearranging, we have:
k^2 - 4 > 0
(k - 2)(k + 2) > 0
To satisfy this inequality, either both factors (k - 2) and (k + 2) must be positive, or both must be negative.
Case 1: (k - 2) > 0 and (k + 2) > 0
This implies that k > 2 and k > -2, which means k must be greater than 2.
Case 2: (k - 2) < 0 and (k + 2) < 0
This implies that k < 2 and k < -2, which means k must be less than -2.
Therefore, the function f(x) = kx^2 - 4x + k has no zeros when k > 2 or k < -2.
Now, let's move to the second part of the question.
To determine the values of k for which the function f(x) = 4x^2 - 3x + 2kx + 1 has two zeros, we need to find the discriminant again.
The quadratic equation is given by ax^2 + bx + c = 0, where in our case, a = 4, b = -3, and c = 2k + 1.
The discriminant (D) is calculated as D = b^2 - 4ac.
Substituting the values, we have D = (-3)^2 - 4(4)(2k + 1).
Simplifying, we get D = 9 - 32(2k + 1).
To have two zeros, the discriminant D must be greater than zero since the quadratic equation intersects the x-axis twice.
So, 9 - 32(2k + 1) > 0.
Simplifying further, we have 9 - 64k - 32 > 0.
Combining like terms, we get -64k - 23 > 0.
Solving for k, we have -64k > 23.
Dividing both sides by -64 (and flipping the inequality sign), we get:
k < 23/64.
Therefore, the values of k for which the function f(x) = 4x^2 - 3x + 2kx + 1 have two zeros are k < 23/64.