Application of DerivativeHard
Question
For which values of ′a′ will the function f(x) = x4 + ax3 + 
+ 1 will be concave upward along the entire real line
+ 1 will be concave upward along the entire real line Options
A.a ∈ [0, ∞)
B.a ∈ (-2, ∞)
C.a ∈ [-2, 2]
D.a ∈ (0, ∞)
Solution
f(x) = x4 + ax3 + 
+ 1
f′(x) = 4x3 + 3ax2 + 3x
f″(x) = 12x2 + 6ax + 3
Now f(x) will be concave upward along the entire
real line iff f″(x) ≥ 0 ∀ x ∈ R
12x2 + 6ax + 3 > 0 ⇒ D ≥ 0
36a2 - 144 ≤ 0
a2 - 4 ≤ 0 ⇒ a ∈ [- 2, 2]
+ 1f′(x) = 4x3 + 3ax2 + 3x
f″(x) = 12x2 + 6ax + 3
Now f(x) will be concave upward along the entire
real line iff f″(x) ≥ 0 ∀ x ∈ R
12x2 + 6ax + 3 > 0 ⇒ D ≥ 0
36a2 - 144 ≤ 0
a2 - 4 ≤ 0 ⇒ a ∈ [- 2, 2]
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