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Let f be a real valued function satisfy f(x) + f(x+4) = f(x+2) + f(x+6) Prove that limit (x to x+8) ∫f(t)dt is constant

Let f be a real valued function satisfy
f(x) + f(x+4) = f(x+2) + f(x+6)
Prove that
limit (x to x+8)
∫f(t)dt is constant

Grade:11

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
7 years ago
Ans:Hello student, please find answer to your question
f(x) + f(x+4) = f(x+2) + f(x+6)…..(1)
f(x+2) + f(x+2+4) = f(x+2+2) + f(x+2+6)
f(x+2) + f(x+6) = f(x+4) + f(x+8)…......(2)
(1) + (2)
f(x) = f(x+8)
I = \int_{x}^{x+8}f(t)dt
I' = f(x+8)-f(x)
I' = 0
\Rightarrow \int_{x}^{x+8}f(t)dt
is constant

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