Question

Assertion :If $f(x+y)+f(x-y)=2f\left(x\right)f\left(y\right)\forall x,y\in R$ and $f\left(0\right)\ne 0$, then $f\left(x\right)$ is an even function. Reason: If $f(-x)=f\left(x\right)$, then $f\left(x\right)$ is even

• 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 $f(x+y)+f(x-y)=2f\left(x\right)f\left(y\right)...\left(i\right)$

Change $x\leftrightarrow y$

$\Rightarrow f(y+x)+f(y-x)=2f\left(y\right)f\left(x\right)...\left(ii\right)$

From $\left(i\right)$ and $\left(ii\right)$ we have

$f(x-y)=f(y-x)....(\ast )$

Putting $y=2x$ then from $(\ast )$ we get $f(-x)=f\left(x\right)$

$\therefore f\left(x\right)$ is even function

So Assertion $\left(A\right)$ & Reason $\left(R\right)$ both are true but Reason $\left(R\right)$ is the correct explanation of Assertion $\left(A\right)$