LimitsHard
Question
If f: R → R be a differentiable function, such that f(x + 2y) = f(x) + f(2y) + 4xy ∀ x, y ∈ R . then
Options
A.f′(1) = f′(0) + 1
B.f′(1) = f′(0) - 1
C.f′(0) = f′(1) + 2
D.f′(0) = f′(1) - 2
Solution
f(x + 2y) = f(x) + f(2y) + 4xy ∀ x, y ∈ R
Replace 2y with y we have
f(x + y) = f(x) + f(y) + 2xy ∀ x, y ∈ R
diff. w.r.t. x
f′(x + y) = f′(x) + 2y
Put x = 1 y = - 1 f′(0) = f′(1) - 2
Replace 2y with y we have
f(x + y) = f(x) + f(y) + 2xy ∀ x, y ∈ R
diff. w.r.t. x
f′(x + y) = f′(x) + 2y
Put x = 1 y = - 1 f′(0) = f′(1) - 2
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