a.maths question (1)

kx^2+(4k-1)x+4k >=0 for all real values of x
means the discriminant is less than or equal to 0.
That is,
D = (4k-1)^2 - 4(k)(4k) <= 0
16k^2 - 8k + 1 - 16k^2 <= 0
-8k +1 <= 0
1 < = 8k
k >= 1/8

For your information, a x^2 + b x + c > 0 for all values of x means the quadratic equation a x^2 + b x + c = 0 has no solution (because a x^2 + b x + c is always > 0 and never = 0).
In additon, a x^2 + b x + c >= 0 for all values of x means the quadratic equation a x^2 + b x + c = 0 has no solution or only a repeated solution. The reason is same as above.

1.find the range of real values of k if kx^2+(4k-1)x+4k>=0 for all real values of x

kx^2+(4k-1)x+4k>=0
B^2-4AC>=0
(4k-1)^2-4*(k)(4k)>=0
16k^2-8k+1-16k^2>=0
1-8k>=0
8k<=1
k<=1/8

So, k<=1/8.