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]
Create a free account to view solution
View Solution FreeMore Application of Derivative Questions
The coordinates of the points on the curve x = a (θ + sin θ), y = a (1−cos θ), where tangent is inc...At any point a curve is equal to-...The point at which the tangent to the curve y = x3 + 5 is perpendicular to the line x + 3y = 2 are-...The curve y-exy + x = 0 has a vertical tangent at -...If the tangent at any point on the curve x4 + y4 = a4 cuts off intercept p and q on the axes, the value of p-4/3 + q-4/3...