Question

Assertion :Let $f:R\to R$ be a function defined by $f\left(x\right)=x^3+x^2+3x+\sin x$, then $f$ 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. Both Assertion and Reason are incorrect

Answer 1

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

Given, $f\left(x\right)=x^3+x^2+3x+\sin x$
$\Rightarrow$ $f^{\prime }\left(x\right)=3x^2+2x+3+\cos x$
$=3{(x+\frac{1}{3})}^2+\frac{8}{3}+\cos x>0$ as $-1\le \cos x\le 1$
$\Rightarrow f\left(x\right)$ is an increasing function (Reason is false)
$\Rightarrow f\left(x\right)$ is one-one function (Assertion is true).