MonotonicityHard
Question
The values of p for which the function f(x) = 
x5 - 3x + ln 5 decreases for all real x is
x5 - 3x + ln 5 decreases for all real x isOptions
A.(- ∞, ∞)
B.
υ (1, ∞)
υ (1, ∞)C.
υ (2, ∞)
υ (2, ∞)D.[1, ∞)
Solution
f(x) = 
x5 - 3x + ln 5
f′(x) =
5x4 - 3 < 0 ∀ x ∈ R
⇒
- 1 ≤ 0
If - 4 ≤ p < 1 then
⇒
≤ 1 - p ⇒ p + 4 ≤ 1 - 2p + p2
⇒ p2 - 3p - 3 ≥ 0
⇒ p ≤
≤ p
⇒ p∈
If p > 1 then
≥ 1 - p
⇒ Always true for p > 1
⇒ p∈
υ (1, ∞)
x5 - 3x + ln 5f′(x) =
5x4 - 3 < 0 ∀ x ∈ R⇒
- 1 ≤ 0If - 4 ≤ p < 1 then
⇒
≤ 1 - p ⇒ p + 4 ≤ 1 - 2p + p2⇒ p2 - 3p - 3 ≥ 0
⇒ p ≤
≤ p⇒ p∈
If p > 1 then
≥ 1 - p⇒ Always true for p > 1
⇒ p∈
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