Final answer:
To find the smallest integer c such that the domain of the function f(x) = (x² + 1)/(x² - xc) is all real numbers, we need to consider the denominator. By setting x(x - c) ≠ 0, we find that c = 1.
Step-by-step explanation:
To find the smallest integer c such that the domain of the function f(x) = (x² + 1)/(x² - xc) is all real numbers, we need to consider the denominator. For the denominator to be defined for all real numbers, it cannot be equal to zero. So, we set x² - xc ≠ 0 and solve for c.
If we factor out an x from the denominator, we get x(x - c) ≠ 0. To ensure that the inequality holds for all real numbers, we need both factors to be nonzero. This means x ≠ 0 and x - c ≠ 0.
Combining the two conditions, we get x ≠ 0 and x ≠ c. Since we want the smallest possible value of c, we choose c = 1. Therefore, the answer is (a) c = 1.