Final answer:
The problem is based on the homogeneity property, requiring us to increase x by the 129th root of 3 to triple the value of f(x), which is not reflected in the provided choices.
Step-by-step explanation:
The exercise in question is dealing with the homogeneity property of functions. We are given that f(x) = cx¹²¹ and we need to find by what factor x must be increased to triple the value of f(x).
The homogeneity property in mathematics states that a function f(x) is homogeneous of degree n if, for any scalar k, f(kx) = k^n * f(x). In our case, we want f(kx) to be three times f(x), so we set up the equation cx¹²¹ * k¹²¹ = 3cx¹²¹.
Simplifying, we find that k¹²¹ = 3, which means k = 3^(1/129). Therefore, x must be increased by the kth factor, which is the 129th root of 3. This is not one of the provided answer choices (a, b, c, d), which suggests those choices might not be relevant to this particular problem.