Let b be a nonzero real number. Suppose f: R R is a differentiable function such that (0)=1. If the derivative f^ of f satisfies the equation f ^ ( x )= f ( x ) b ^2+ x ^2 for all x R, then which of the following statements is/are TRUE? (A) If b>0, then f is an increasing function (B) If b<0, then f is a decreasing function (C) (x)(-x)=1 for all x R (D) (x)-f(-x)=0 for all x R

Let b be a nonzero real number. Suppose f: R R is a differentiabl…

MCQ

Let $b$ be a nonzero real number. Suppose $f: R \rightarrow R$ is a differentiable function such that $(0)=1$.

If the derivative $f^{\prime}$ of $f$ satisfies the equation $f ^{\prime}( x )=\frac{ f ( x )}{ b ^2+ x ^2}$ for all $x \in R$, then which of the following statements is/are TRUE?

$(A)$ If $b>0$, then $f$ is an increasing function

$(B)$ If $b<0$, then $f$ is a decreasing function

$(C)$ $(x)(-x)=1$ for all $x \in R$

$(D)$ $(x)-f(-x)=0$ for all $x \in R$

A
$A,B$
B
$A,D$
C
$B,C$
✓
$A,C$

✓

Answer

Correct option: D.

$A,C$

d
$f^{\prime}( x )=\frac{f( x )}{ b ^2+ x ^2}$

$\int \frac{f^{\prime}( x )}{f( x )} dx =\int \frac{ dx }{ x ^2+ b ^2}$

$\Rightarrow \ell n |f( x )|=\frac{1}{ b } \tan ^{-1}\left(\frac{ x }{ b }\right)+ c$

Now $f(0)=1$

$\therefore c=0$

$\therefore|f( x )|= e ^{\frac{1}{b} \cdot \tan ^{-1}\left(\frac{x}{b}\right)}$

$\Rightarrow f( x )= \pm e ^{\frac{1}{ b } \tan ^{-1}\left(\frac{x}{ b }\right)}$

since $f(0)=1 \therefore \quad f( x )= e ^{\frac{1}{b} \tan ^{-1}\left(\frac{x}{b}\right)}$

$x \rightarrow- x$

$f(- x )= e ^{-\frac{1}{ b } \tan ^{-1}\left(\frac{ x }{ b }\right)}$

$\therefore f( x ) \cdot f(- x )= e ^0=1 \text { (option } C \text { ) }$

and for $b>0$

$f( x )= e ^{\frac{1}{6} \operatorname{tax}-1\left(\frac{x}{6}\right)}$

$\Rightarrow f( x )$ is increasing for all $x \in R$ (option $A$ )

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from STD 12 - 6. Application of derivatives→STD 12 - 6. Application of derivatives — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

If $f(x) = \int_a^x {{t^3}{e^t}\,dt\,,} $ then $\frac{d}{{dx}}\,f(x) = $

→

If $A = \left[ {\begin{array}{*{20}{c}}1&{ - 1}\\2&{ - 1}\end{array}} \right],\,\,B = \left[ {\begin{array}{*{20}{c}}a&1\\b&{ - 1}\end{array}} \right]$ and ${(A + B)^2} = {A^2} + {B^2}$, then the value of $a$ and $b$ are

→

Area bounded by the lines $y = |x| - 2$ and $y = 1 - |x - 1|$ is equal to:

→

Consider the system of equations $\mathrm{ax}+\mathrm{by}=0, \mathrm{cx}+\mathrm{dy}=0$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \in\{0,1\}$.

$STATEMENT -1$ : The probability that the system of equations has a unique solution is $3 / 8$. and $STATEMENT - 2$: The probability that the system of equations has a solution is $1$ .

→

If $f:R \to R$ and $g:R \to R$ are given by $f(x) = \;|x|$ and $g(x) = \;|x|$ for each $x \in R$, then $\{ x \in R\;:g(f(x)) \le f(g(x))\} = $

→

If the function $f(x) = \left\{ \begin{array}{l}1 + \sin \frac{{\pi x}}{2}\,\,,\,{\rm{\,\,for}}\,\, - \infty < x \le 1\\\,\,\,\,\,\,\,\,ax + b,\,{\rm{\,\,for}}\,\,1 < x < 3\\\,\,\,\,6\tan \frac{{x\pi }}{{12}},\,{\rm{\,\,for\,\,}}3 \le x < 6\end{array} \right.$ is continuous in the interval $( - \infty ,\,6)$, then the values of $a$ and $b$ are respectively

→

Consider the function $f (x) = x^3 - 8x^2 + 20x -13$
Area enclosed by $y = f (x)$ and the co-ordinate axes is

→

In the figure, a vector $x$ satisfies the equation $x - w = v$. Then $x =$