Question

Assertion :Consider the function $f\left(x\right)$ satisfying the relation $f(x+1)+f(x+7)=0\forall x\in R.$
The possible least value of $t$ for which ${\int }_a^{a+t}f\left(x\right)dx$ is independent of $a$ is $12$ Reason: $f\left(x\right)$ is a periodic function.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given $f(x+1)+f(x+7)=0\because x\in R$
Replacing $x$ by $x-1,$ we have $f\left(x\right)+f(x+6)=0$ ........(1)
Now, replacing $x$ by $x+6,$ we have $f(x+6)+f(x+12)=0$ ........(2)
From equations (1) and (2), we have $f\left(x\right)=f(x+12)$ ........(3)
Hence, $f\left(x\right)$ is periodic with period $12$.
Thus, ${\int }_a^{a+t}f\left(x\right)dx$ is independent of a if $t$ is positive integral multiple of $12$.
Then possible value of $t$ is $12$.