Question

Assertion :Let $f:R\to R$, $f\left(x\right)=x^3+x^2+100x+5\sin x$, then $f\left(x\right)$ is bijective. Reason: $3x^2+2x+95>0x\in R$.

• 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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

We have,
$f\left(x\right)=x^3+x^2+100x+5\sin x$
Since $-1\le \sin x\le 1$ and $-\infty <x^3+x^2+100x<\infty$
Range of $f$ is $R$ which is equal to co-domain $\Rightarrow$ $f$ is onto function.
Now $f^{\prime }\left(x\right)=3x^2+2x+100+5\cos x$
Also $-1\le \cos x\le 1$
$\Rightarrow f^{\prime }\left(x\right)=3x^2+2x+100+5(-1)=3x^2+2x+95$
Discriminant of above quadratic is $4-4\left(3\right)\left(95\right)<0$
which means $f^{\prime }\left(x\right)>0\forall x\in R\Rightarrow f$ is increasing function $\Rightarrow$ $f$ is an one-one function
Hence both statements are correct but Reason is not correct explanation of Assertion.