Consider two functions of time $(t)$, $$f(t) = 0.01\ t^2$$ $$g(t) = 4\ t$$ where $0 < t < \infty$. Now consider the following two statements: (i) For some $t > 0$, $g(t) > f(t)$. (ii) There exists a $T$, such that $f(t) > g(t)$ for all $t > T$. Which one of the following options is TRUE?

We are given the two functions for $t > 0$:

1. Evaluating Statement (i): We determine the range of $t$ for which $g(t) > f(t)$:

Since $t > 0$, we can divide both sides by $t$:

Thus, for all $t \in (0, 400)$, we have $g(t) > f(t)$. For example, at $t = 1$, $g(1) = 4$ and $f(1) = 0.01$, so $g(1) > f(1)$. Hence, statement (i) is correct.

2. Evaluating Statement (ii):We find when $f(t) > g(t)$:

Dividing both sides by $t$ (since $t > 0$):

Choosing $T = 400$, for every $t > T$, $f(t) > g(t)$ holds true. Hence, statement (ii) is correct.

Therefore, both (i) and (ii) are correct, which corresponds to (C).