Consider the functions I. \(e^{-x}\) II. \(x^{2}-\sin x\) III.…
2020
Consider the functions
I. \(e^{-x}\)
II. \(x^{2}-\sin x\)
III. \(\sqrt{x^{3}+1}\)
Which of the above functions is/are increasing everywhere in [0, 1] ?
- A.
Ⅲ only
- B.
Ⅱ only
- C.
Ⅱ and Ⅲ only
- D.
Ⅰ and Ⅲ only
Attempted by 57 students.
Show answer & explanation
Correct answer: A
Key idea: test each function's derivative on [0,1].
For e^{-x}: derivative is -e^{-x} < 0 on [0,1], so this function is decreasing (not increasing).
For x^2 - sin x: derivative is 2x - cos x. At x = 0 this equals -1, so the derivative is negative near 0 and the function is not increasing everywhere on [0,1].
For √(x^3+1): derivative is (3x^2)/(2√(x^3+1)) ≥ 0 for x in [0,1], with strict positivity for x > 0, so the function is monotone increasing on [0,1].
Conclusion: Only √(x^3+1) is increasing everywhere on [0,1].