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Consider two functions of time (t), f(t) = 0.01 t² and g(t) = 4 t, where 0 < t < ∞. 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?

1. only (i) is correct
2. only (ii) is correct
3. both (i) and (ii) are correct
4. neither (i) nor (ii) is correct

Correct answer: both (i) and (ii) are correct

Solution

Both statements are correct because for small values of t, g(t) can exceed f(t), but as t increases, the quadratic growth of f(t) eventually surpasses the linear growth of g(t), establishing a point T beyond which f(t) is always greater than g(t).

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