RenB wrote:
If x and y are integers such that 2 < x ≤ 8 and 2 < y ≤ 9, what is the maximum value of (1/x - x/y)?
A. \(-3\frac{1}{8}\)
B. 0
C. 1/5
D. 5/18
E. 2
Hi,
I did get the logic that the value of 1/x has to be maximised and value of x/y minimized
However, I got confused seeing the x in the numerator of x/y. I thought this numerator x also has to be factored, even if it implies reducing the value of 1/x
I am still unclear about the logic or impact of x in the numerator of x/y
Can you please help me out with the logic here and correct my line of thought?
Looking forward to hearing from you!
To maximize (1/x - x/y), we need to maximize 1/x and minimize x/y.
To maximize 1/x, we need to minimize x. The minimum possible value of x is 3. Thus, the maximum value of 1/x is 1/3.
To minimize x/y, we need to minimize x (AGAIN) and maximize y. The minimum possible value of x is 3 and the maximum possible value of y is 9. Thus, the minimum value of x/y is 3/9.
Therefore, the maximum value of (1/x - x/y) is obtained when x = 3 and y = 9, which is 1/3 - 3/9 = 0.
Answer: B.
P.S. To address your doubt, when maximizing (1/x - x/y), the goals of minimizing x for the first term (1/x) and minimizing x for the second term (x/y) are aligned and do not contradict each other. By minimizing x, we are able to maximize the overall expression, as both terms benefit from the same action.