Question

Assertion :Statement-1: $e^x+e^{-x}>2+x^2,x\ne 0$ Reason: Statement-2: $f\left(x\right)=e^x+e^{-x}-2-x^2$ is an increasing 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 C Assertion is correct but Reason is incorrect

We know that

$AM\ge GM$

$\frac{e^x+e^{-x}}{2}\ge 1$

$e^x+e^{-x}\ge 2$

$\because x^2>0$

So, $e^x+e^{-x}>2+x^2$

Thus, the assertion is correct.

$f\left(x\right)=e^x+e^{-x}-2-x^2$

$f(-x)=e^{-x}+e^x-2-x^2$

$\because f\left(x\right)=f(-x)$

i.e. function is symmetric about y-axis

So it can not be increasing function.

Thus, the reason 2 is incorrect.