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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

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

Vedclass · Answer

(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.

Similar Questions

$f(x)=\frac{x}{e^x-1}+\frac{x}{2}+2 \cos ^3 \frac{x}{2}$ on $R-\{0\}$ is

$A = \{1, 2, 3, 4\}$ and $B = \{1, 2, 3, 4, 5, 6\}$ are two sets, and the function $f: A \rightarrow B$ is defined by $f(x) = x + 2$ for all $x \in A$. Then the function $f$ is:

If a set $A$ has $m$ elements and set $B$ has $n$ elements and the number of injections from $A$ to $B$ is $2520$. Then,$m$ is equal to

Let $Z$ denote the set of integers. Define $f: Z \rightarrow Z$ by $f(x) = \begin{cases} \frac{x}{2}, & x \text{ is even} \\ 0, & x \text{ is odd} \end{cases}$. Then $f$ is:

Let $f: N \rightarrow N$ be defined by $f(n) = \begin{cases} \frac{n+1}{2}; & \text{if } n \text{ is odd} \\ \frac{n}{2}; & \text{if } n \text{ is even} \end{cases}$, for all $n \in N$ then $f$ is $\dots \dots \dots$

Vedclass Test Series

Exam Paper Generator

Online Exam Module

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.