Question

Consider two functions of time $(t)$, \[ f(t) = 0.01t^2, g(t) = 4t \] 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?

• A. only (i) is correct
• B. only (ii) is correct
• C. both (i) and (ii) are correct
• D. neither (i) nor (ii) is correct

Hint:

When comparing growth rates, linear functions dominate quadratics for small $t$, but quadratics eventually dominate for large $t$. Always check both small and large values.

Answer 1

The Correct Option is C

Step 1: Compare $f(t)$ and $g(t)$.
We want to solve when $f(t)>g(t)$ or $g(t)>f(t)$. \[ f(t) = 0.01t^2, g(t) = 4t \] So we compare: \[ 0.01t^2 \text{vs.} 4t \] Step 2: Inequality setup.
Check when $g(t)>f(t)$: \[ 4t>0.01t^2 \Rightarrow 0.01t^2 - 4t<0 \] Factor: \[ 0.01t(t - 400)<0 \] Step 3: Range for inequality.
This inequality is true when $0<t<400$.
Thus, for some positive $t$ (specifically, $t<400$), $g(t)>f(t)$.
Hence, statement (i) is correct.
Step 4: Check for large $t$.
As $t \to \infty$, the quadratic term $0.01t^2$ grows faster than the linear term $4t$.
Thus, beyond $t = 400$, \[ f(t)>g(t) \forall t>400 \] So, statement (ii) is also correct. Step 5: Conclusion.
Both (i) and (ii) are true. \[ \boxed{\text{Both (i) and (ii) are correct.}} \]