QJEEM21Maf690: JEE Main 2021 Mathematics - Class 12 JEEM21Maf690 Solution: Let f : R R be a continuous function such that f(x) + f(x + 1)…

Question

Verified Solution • 2025–26 Session moderate

Let f : R

\to

R be a continuous function such that f(x) + f(x + 1) = 2, for all x

\in

R.

If

{I_1} = \int\limits_0^8 {f(x)dx}

and

{I_2} = \int\limits_{ - 1}^3 {f(x)dx}

, then the value of I₁ + 2I₂ is equal to ____________.

Accepted Answer

f(x) + f(x + 1) = 2

.... (i)

x \to (x + 1)

f(x + 1) + f(x + 2) = 2

.... (ii)

by (i) & (ii)

f(x) - f(x + 2) = 0

f(x + 2) = f(x)

\therefore

f(x) is periodic with T = 2

{I_1} = \int_0^{2 \times 4} {f(x)dx} = 4\int_0^2 {f(x)dx}

{I_2} = \int_{ - 1}^3 {f(x)dx} = \int_0^4 {f(x + 1)dx} = \int_0^4 {(2 - f(x))dx}

\Rightarrow

{I_2} = 8 - 2\int_0^2 {f(x)dx}

\Rightarrow

{I_2} = 8 -

{{{I_1}} \over 2}

\Rightarrow

{I_1} + 2{I_2} = 16

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