Consider two functions of time (π‘), π(π‘) = 0.01 π‘2 π(π‘) = 4π‘ where 0 <β¦
2023
Consider two functions of time (π‘),
Β Β Β Β Β Β Β Β Β Β Β Β π(π‘) = 0.01 π‘2
Β Β Β Β Β Β Β Β Β Β Β Β π(π‘) = 4π‘
where 0 < π‘ < β.
Now consider the following two statements:
(i) For some π‘ > 0, π(π‘) > π(π‘).
(ii) There exists a π, such that π(π‘) > π(π‘) for all π‘ > π.
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
Attempted by 37 students.
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Correct answer: C
Answer: both statements are true.
Reasoning:
Consider the inequality for the first statement: 4t > 0.01 tΒ². For t > 0 we can divide by t to get 4 > 0.01 t, so 0 < t < 400. Thus there are many positive values of t (for example t = 1) for which 4t > 0.01 tΒ², so the first statement is true.
For the second statement, solve 0.01 tΒ² > 4t. For t > 0 divide by t to get 0.01 t > 4, so t > 400. Therefore choosing T = 400 ensures that for every t > T we have 0.01 tΒ² > 4t, proving the second statement is true.
The two functions are equal at t = 400 (since 0.01Β·400Β² = 4Β·400). For t < 400 the linear function is larger; for t > 400 the quadratic is larger. Hence both statements hold.