Set, Relation and FunctionHard
Question
f(x) and g(x) are two differentiable functions on [0, 2] such that f″(x) - g″(x) = 0 f′(1) = 2g′(1) = 4f(2) = 3g(2) = 9 then f(x) - g(x) at x = 3/2 is
Options
A.0
B.2
C.10
D.5
Solution
∵ f″(x) - g″(x) = 0
Integrating, f′(x) - g′(x) = c ⇒ f′(1) - g′(1) = c ⇒ 4 - 2 = c ⇒ c = 2
∴ f′(x) - g′(x) = 2 ; Integrating, f(x) - g(x) = 2x + C1
⇒ f(2) - g(2) = 4 + c1 ⇒ 9 - 3 = 4 + c1 ⇒ c1 = 2 ∴ f(x) - g(x) = 2x + 2
At x = 3/2, f(x) - g(x) = 3 + 2 = 5
Integrating, f′(x) - g′(x) = c ⇒ f′(1) - g′(1) = c ⇒ 4 - 2 = c ⇒ c = 2
∴ f′(x) - g′(x) = 2 ; Integrating, f(x) - g(x) = 2x + C1
⇒ f(2) - g(2) = 4 + c1 ⇒ 9 - 3 = 4 + c1 ⇒ c1 = 2 ∴ f(x) - g(x) = 2x + 2
At x = 3/2, f(x) - g(x) = 3 + 2 = 5
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