Statement-I: Let f : R rightarrow R be a func | vedclass

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(C) To check if $f(x) = x^3 + x^2 + 3x + \sin x$ is one-one,we find its derivative $f'(x)$.$f'(x) = 3x^2 + 2x + 3 + \cos x$.We know that $-1 \leq \cos

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Statement-$I$ is true,Statement-$II$ is false.

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(C) To check if $f(x) = x^3 + x^2 + 3x + \sin x$ is one-one,we find its derivative $f'(x)$.$f'(x) = 3x^2 + 2x + 3 + \cos x$.We know that $-1 \leq \cos x \leq 1$.So,$f'(x) \geq 3x^2 + 2x + 3 - 1 = 3x^2 + 2x + 2$.For the quadratic $3x^2 + 2x + 2$,the discriminant $D = b^2 - 4ac = (2)^2 - 4(3)(2) = 4 - 24 = -20 Since the coefficient of $x^2$ is positive $(3 > 0)$ and $D  0$ for all $x \in R$.Thus,$f'(x) > 0$ for all $x \in R$.Since $f'(x) > 0$,$f(x)$ is a strictly increasing function.$A$ strictly increasing function is always a one-one function.Therefore,Statement-$I$ is true.Statement-$II$ claims $f(x)$ is a decreasing function,which is false because $f'(x) > 0$ implies it is increasing.

Statement-$I$: Let $f : R \rightarrow R$ be a function such that $f(x) = x^3 + x^2 + 3x + \sin x$. Then $f$ is a one-one function.
Statement-$II$: $f(x)$ is a decreasing function.

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Statement-$I$: Let $f : R \rightarrow R$ be a function such that $f(x) = x^3 + x^2 + 3x + \sin x$. Then $f$ is a one-one function.Statement-$II$: $f(x)$ is a decreasing function.