how would i determine the value of k which the maximum value of the function is an integer if y= -4x^2 + kx - 1.

Question

directrix · Answer

I am thinking k = 4 or -4.
 That comes from the discriminant k^2 - 4(-4)(-1) = k^2 - 16. 
y = -4x^2 + 4x -1 which has max of 0 at x = 1/2.
Also, y = -4x^2 -4x - 1 which also has max of 0  at x = -1/2.
Nothing conclusive yet, just thinking.

anonymous · Answer

the answer given at the back of the book is that k must be divisible by four

directrix · Answer

I set k = 24 based on your post. y = -4x^2 + 24x - 1 now has max at 35. 
Upon initial reading of the question, I thought that the maximum value was a set number for the problem. Now, I see that the only restriction on the maximum is that it is an integer.