Question

Assertion :

Let $f:R\to R,f\left(x\right)=x^3+x^2+3x+\sin x.$

$f\left(x\right)$ is one-one. Reason: $f\left(x\right)$ is decreasing 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. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

$f\left(x\right)=x^3+x^2+3x+\sin x$
$\Rightarrow f^{\prime }\left(x\right)=3x^2+2x+3+\cos x$
Now we know, $-1\le \cos x\le 1$
$\Rightarrow f^{\prime }\left(x\right)\ge 3x^2+2x+3-1=3x^2+2x+2$
Now discriminant of quadratic $3x^2+2x+2$ is $2^2-4\left(3\right)\left(2\right)<0$
Hence $f^{\prime }\left(x\right)>0\forall x\in R$
Thus $f$ is increasing function $\Rightarrow$ $f$ is one-one
Therefore Assertion is correct but Reason is incorrect.