Consider the following statements about the polynomial p(x), where a < b:
I p(a) ≤ p(b)
II p'(a) ≤ p'(b)
III p''(a) ≤ p''(b)
Which of these statements is a necessary condition for p(x) to be increasing for a ≤ x ≤ b?
- Anone of them
- BI only
- CII only
- DIII only
- EI and II only
- FI and III only
- GII and III only
- HI, II and III
Show the answer and worked solution
answer · B
- Anone of them
- BI only
- CII only
- DIII only
- EI and II only
- FI and III only
- GII and III only
- HI, II and III
"Increasing on [a, b]" is a statement about the function's values, so I follows at once: if the graph rises across the interval then p(a) ≤ p(b). The tempting slip is to think the derivative must also rise, but increasing says only that p' stays non-negative, not that it grows. Take p(x) = 3x − x3 on [0, 12]: here p' = 3 − 3x2 > 0 so p is increasing, yet p'(0) = 3 is bigger than p'!(12) = 2.25, and p'' = −6x is also decreasing. So II and III both fail, leaving I only.