Final answer:
The smallest integer value that f(7) can be, given the conditions f(4)=6 and f²(x) ≤ 3 for 4 ≤ x ≤ 7, is 0. This is because f(x) must be non-negative and its square does not exceed 3, the closest integer that meets these criteria is 0.
Step-by-step explanation:
The student's question involves determining the smallest possible value of f(7) given the function conditions. Given f(4)=6 and f²(x) ≤ 3 for the interval 4 ≤ x ≤ 7, we can infer that the function value must be non-negative since the square of a real number is always non-negative.
The condition f²(x) ≤ 3 implies that f(x) itself must be less than or equal to √3, because if it were larger, its square would exceed 3. Therefore, the function cannot take on a value greater than √3 in the given interval. Knowing that the square root of 3 is approximately 1.732, the function f(x) may range down to -1.732 while satisfying the condition f²(x) ≤ 3. However, since we are seeking the smallest integer value that f(7) can possibly be, and f(x) must also be non-negative, the smallest integer value for f(7) is 0.