Final answer:
The function f(x) = 300x² + x log₃(x) + 2ˣ⁴⁰⁰ is increasing across its domain wherever x > 0. The major contributing components of the function, which are quadratic, logarithmic, and exponential, all have positive derivatives for x > 0, leading to the overall function increasing for positive x values.
Step-by-step explanation:
To determine where the function f(x) = 300x² + x log₃(x) + 2ˣ⁴⁰⁰ is increasing or decreasing, it's necessary to look at its derivative since the sign of the first derivative can indicate whether a function is increasing (positive derivative) or decreasing (negative derivative). Because the function contains quadratic, logarithmic, and exponential components, all of which are individually increasing over the domain of positive numbers, the overall derivative is likely to be positive for x > 0 as well.
Moreover, since the quadratic term 300x² dominates as x becomes very large, and because the derivative of this term is 600x which is positive for x > 0, and similarly the derivatives of x log₃(x) and 2ˣ⁴⁰⁰ are also positive for x > 0. However, log(x) is undefined for x <= 0, and thus, we won't consider decreasing in negative x. Consequently, without calculating the exact derivative but analyzing the behavior of the derivative's components, we can infer the function is increasing everywhere in its domain of positive x.