Question

If y = -4x^2 + kx - 1, determine the value(s) of k for which the maximum value of the function is an integer. Explain your reasoning.

Answer 1

First, take the derivative. Then set the derivative equal to 0 to find the critical points. Since your function is a parabola, there will be only 1 critical point.

Answer 2

Namely, your derivative will be \(-8x+k\) so if you set that equal to zero you get \(k=8x\) or \(x=k/8\).

Answer 3

If you plug that value back into your equation, you find that \[y=-4({k\over8})^2 +k({k\over8}) -1\] so \[y=-{k^2 \over 16}+{k^2 \over 8}-1\]so to be an integer, 16 has to divide \(k^2\). In other words, 4 divides \(k\). So \(k=4a\) for \(a\in\mathbb{Z}\)