tar013 wrote:For all non-negative integers x and n such that 0 ≤ x ≤ n, the function f(x,n) is defined by the equation f(x,n) = x^(n-x). The smallest value of n for which the maximum of f(x,n) occurs when x = 4 is
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
The portions in red reflect what I believe is intended.
We can plug in the answers, which represent the smallest value of n for which the maximum of f(x,n) occurs when x = 4.
Since we need the SMALLEST value of n, we should start with the smallest answer choice.
Answer choice A: n=6
When x=0, f(x,n) = x^(n-x) = 0^(6-0) =
0^6 = 0.
When x=1, f(x,n) = x^(n-x) = 1^(6-1) =
1^5 = 1.
When x=2, f(x,n) = x^(n-x) = 2^(6-2) =
2^4 = 16.
When x=3, f(x,n) = x^(n-x) = 3^(6-3) =
3^3 = 27.
When x=4, f(x,n) = x^(n-x) = 4^(6-4) =
4^2 = 16.
When x=5, f(x,n) = x^(n-x) = 5^(6-5) =
5^1 = 5.
When x=6, f(x,n) = x^(n-x) = 6^(6-6) =
6^0 = 1.
The maximum value does not occur when x=4.
Eliminate A.
The values of the function have been highlighted above in red.
They exhibit the following pattern:
The bases are equal to the consecutive integers between 0 and n, inclusive, in ASCENDING order.
When x=4, the base is equal to 4.
The powers are equal to the consecutive integers between n and 0, inclusive, in DESCENDING order.
For the remaining answer choices, we can follow this pattern to determine whether the maximum value occurs when the base is equal to 4.
Answer choice B: n=7
The bases will be 0, 1, 2, 3, 4, 5, 6, 7.
The powers will be 7, 6, 5, 4, 3, 2, 1, 0.
Thus, the values of the function will be:
0�, 1�, 2�, 3�, 4³, 5², 6¹, 7�.
The greatest value here is not 4³.
Eliminate B.
Answer choice C: n=8
The bases will be 0, 1, 2, 3, 4, 5, 6, 7, 8.
The powers will be 8, 7, 6, 5, 4, 3, 2, 1, 0.
Thus, the values of the function will be:
0�, 1�, 2�, 3�, 4�, 5³, 6², 7¹, 8�.
The greatest value here is 4�.
The correct answer is
C.
