Range of Algebraic Expression

Let $k = \frac{x}{y}$. Then $c = 16k + \frac{49}{k}$.

• If $k > 0$, by AM-GM: $c \ge 2\sqrt{16k \cdot \frac{49}{k}} = 2 \times 4 \times 7 = 56$
• If $k < 0$, let $k = -t$ ($t > 0$): $c = -\left(16t + \frac{49}{t}\right) \le -56$

Thus, $c$ can take any value in $(-\infty, -56] \cup [56, \infty)$. Therefore, $c$ cannot take any value in the interval $(-56, 56)$.

Among the options:

• $-70 \le -56$ (valid)
• $60 \ge 56$ (valid)
• $-60 \le -56$ (valid)
• $-50$ lies in $(-56, 56)$ (invalid)

Hence, $c$ cannot take the value $-50$.