Question

Assertion :Let $f:R\to R$ be a function such that $f\left(x\right)=x^3+x^2+3x+\sin x$. Then, $f$ is one-one. Reason: $f\left(x\right)$ is decreasing function

• A. Assertion and Reason are correct and the reason is the correct explanation for the assertion.
• B. Assertion and Reason are correct and the reason is Not the correct explanation for the assertion.
• C. Assertion is correct while the Reason is incorrect.
• D. Assertion is incorrect while the Reason is correct.

Answer 1

The correct option is C Assertion is correct while the Reason is incorrect.

$f\left(x\right)=(x^3+3x)+x^2+\sin x$

Now $x^3+3x$ is itself a one-one function since $y_1=y_2$ implies $x_1=x_2$

And $\sin \left(x\right)$ is a monotonic function one-one in the interval $[\frac{-\pi }{2},\frac{\pi }{2}]$.

Hence $f\left(x\right)$ is also a one one function.

$f^{\prime }\left(x\right)$ $=3x^2+3+2x+\cos x$
$=(3x^2+2x+3)+\cos x$

Now consider $h\left(x\right)=3x^2+2x+3$
D $=4-36=-32$

Hence $D<0$
Or $h\left(x\right)>0$ for all $x.$
And $\cos \left(x\right)$ is a monotonic function.

Hence $f\left(x\right)$ is not a decreasing function.