Application of DerivativeHard
Question
Let f(x) be a nonzero function whose all successive derivative exist and are nonzero. If f(x), f′(x) and f″(x) are in G.P. and f(0) = 1, f′(0) = 1, then -
Options
A.f′(x) < 0 ∀ x ∈ R
B.f″(x) < 1 ∀ x ∈ R
C.f″(0) ≠ f″′(0)
D.f″(x) > 0 ∀ x ∈ R
Solution
(f′(x))2 = f(x)f″(x)
⇒ (y′) 2 = yy″
⇒
dx ⇒ ln y = ln y′ + c1
f(0) =1, f′(0) =1 ⇒ c1 = 0
∴ y = y′
⇒
1 dx ⇒ ln y = x + c2
f (0) = 1 c2 = 0
∴ y = ex
y″ > 0 ∀ x ∈ R
⇒ (y′) 2 = yy″
⇒
dx ⇒ ln y = ln y′ + c1f(0) =1, f′(0) =1 ⇒ c1 = 0
∴ y = y′
⇒
1 dx ⇒ ln y = x + c2f (0) = 1 c2 = 0
∴ y = ex
y″ > 0 ∀ x ∈ R
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