Discussion
Consider two functions of time \((t)\),
\[ f(t) = 0.01\, t^{2} \]
\[ g(t) = 4\, t \]
where \(0 \lt t \lt \infty\).
Now consider the following two statements:
(i) For some \(t \gt 0\), \(g(t) \gt f(t)\).
(ii) There exists a \(T\), such that \(f(t) \gt g(t)\) for all \(t \gt T\).
Which one of the following options is TRUE?
Answer: C
Both (i) and (ii) are correct.
Compare the two functions by their difference:
\[ \begin{aligned} g(t) - f(t) &= 4t - 0.01t^{2} \\ &= 0.01t\,(400 - t) \end{aligned} \]
For \(0 \lt t \lt 400\) the difference is positive, so \(g(t) \gt f(t)\); for example at t = 1, g = 4 while f = 0.01. That makes statement (i) true.
For \(t \gt 400\) the difference is negative, so taking T = 400 gives \(f(t) \gt g(t)\) for all \(t \gt T\). Statement (ii) is also true: a quadratic with a positive leading coefficient eventually outgrows any linear function, however small its coefficient. The options naming only one statement, or neither, are therefore wrong.
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