If the maximum value of a, for which the function f_ a (x)= ^ -1 … - Vidyadip

MCQ

If the maximum value of $a$, for which the function $f_{a}(x)=\tan ^{-1} 2 x-3 a x+7$ is non-decreasing in $\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)$, is $\bar{a}$, then $f_{a}\left(\frac{\pi}{8}\right)$ is equal to

A
$8-\frac{\pi}{4}$
B
$8-\frac{4 \pi}{9\left(4+\pi^{2}\right)}$
C
$8\left(\frac{1+\pi^{2}}{9+\pi^{2}}\right)$
✓
$8-\frac{9 \pi}{4\left(9+\pi^{2}\right)}$

✓

Answer

Correct option: D.

$8-\frac{9 \pi}{4\left(9+\pi^{2}\right)}$

d
$Bonus$

$f _{ a }( x )=\tan ^{-1} 2 x -3 ax +7$

$f _{ a }^{\prime}( x )=\frac{2}{1+4 x ^{2}}-3 a \geq 0$

$a \leq\left(\frac{2}{3\left(1+4 x ^{2}\right)}\right)_{\text {min. }}$ at $x =\pm \frac{\pi}{6}$

$a _{\max }=\overline{ a }=\frac{6}{9+\pi^{2}}$

$f _{ a }\left(\frac{\pi}{8}\right)=\tan ^{-1} \frac{\pi}{4}-3 \frac{6}{9+\pi^{2}} \frac{\pi}{8}+7=\tan ^{-1} \frac{\pi}{4}-\frac{9 \pi}{4\left(\pi^{2}+9\right)}+7$

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If the circle ${x^2} + {y^2} = {a^2}$ cuts off a chord of length $2b$ from the line $y = mx + c$, then

The sum of all the products of the first $n$ natural numbers taken two at a time is

The locus of the vertices of the family of parabolas $y = \frac{{{a^3}{x^2}}}{3} + \frac{{{a^2}x}}{2} - 2a$ is

The distance of the point $Q (0,2,-2)$ form the line passing through the point $P (5,-4,3)$ and perpendicular to the lines $\overrightarrow{ r }=(-3 \hat{ i }+2 \hat{ k })+\lambda(2 \hat{ i }+3 \hat{ j }+5 \hat{ k }), \lambda \in R $ and $ \overrightarrow{ r }=(\hat{ i }-2 \hat{ j }+\hat{ k })+ \mu(-\hat{i}+3 \hat{j}+2 \hat{k}), \mu \in R$ is

Let $\mathrm{C}$ be the centroid of the triangle with vertices $(3,-1),(1,3)$ and $(2,4) .$ Let $P$ be the point of intersection of the lines $x+3 y-1=0$ and $3 \mathrm{x}-\mathrm{y}+1=0 .$ Then the line passing through the points $\mathrm{C}$ and $\mathrm{P}$ also passes through the point

When $9^{th}$ term of $A.P$ is divided by its $2^{nd}$ term then quotient is $5$ and when $13^{th}$ term is divided by $6^{th}$ term then quotient is $2$ and Remainder is $5$ then find first term of $A.P.$

${d \over {dx}}{\tan ^{ - 1}}{{4\sqrt x } \over {1 - 4x}} = $

If $i + 2j + 3k$ and $3i - 2j + k$ represents the adjacent sides of a parallelogram, then the area of this parallelogram is

If $\omega $ is the cube root of unity, then $\left| {\begin{array}{*{20}{c}}1&\omega &{{\omega ^2}}\\\omega &{{\omega ^2}}&1\\{{\omega ^2}}&1&\omega \end{array}} \right|$=

The ends of the base of an isosceles triangle are at $(2a,\;0)$ and $(0,\;a).$ The equation of one side is $x=2a$ The equation of the other side is