The function, f(x)=(3 x-7) x^ 2 / 3 , x R, is increasing for all … - Vidyadip

MCQ

The function, $f(x)=(3 x-7) x^{2 / 3}, x \in R,$ is increasing for all $x$ lying in

A
$(-\infty, 0) \cup\left(\frac{3}{7}, \infty\right)$
✓
$(-\infty, 0) \cup\left(\frac{14}{15}, \infty\right)$
C
$\left(-\infty, \frac{14}{15}\right)$
D
$\left(-\infty,-\frac{14}{15}\right) \cup(0, \infty)$

✓

Answer

Correct option: B.

$(-\infty, 0) \cup\left(\frac{14}{15}, \infty\right)$

b
$f(x)=(3 x-7) x^{2 / 3}$

$\Rightarrow \quad f(x)=3 x^{5 / 3}-7 x^{2 / 3}$

$\Rightarrow \quad f^{\prime}(x)=5 x^{2 / 3}-\frac{14}{3 x^{1 / 3}}$

$=\frac{15 x-14}{3 x^{1 / 3}}>0$

$\therefore \quad f ^{\prime}( x )>0 \forall x \in(-\infty, 0) \cup\left(\frac{14}{15}, \infty\right)$

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

${x^2} = xy$ is a relation which is

An anti-aircraft gun take a maximum of four shots at an enemy plane moving away from it. The probability of hitting the plane at the first, second, third and fourth shot are $0.4, 0.3, 0.2$ and $0.1$ respectively. The probability that the gun hits the plane is

The coefficients of three successive terms in the expansion of ${(1 + x)^n}$ are $165, 330$ and $462$ respectively, then the value of n will be

An ellipse inscribed in a semi-circle touches the circular arc at two distinct points and also touches the bounding diameter. Its major axis is parallel to the bounding diameter. When the ellipse has the maximum possible area, its eccentricity is

Let $R$ be a rectangle, $C$ be a circle, and $T$ be a triangle in the plane. The maximum possible number of points common to the perimeters of $R, C$ and $T$ is

The solution of the differential equation $\frac{{dy}}{{dx}} + \frac{{1 + {x^2}}}{x} = 0$ is

If $\int\limits_0^{\frac{\pi }{3}} {\frac{{\tan \,\,\theta }}{{\sqrt {2k\,\sec \,\theta } }}} \,d\theta \, = \,1 - \frac{1}{{\sqrt 2 }},(k > 0),$ then the value of $k$ is

If the ratio of the roots of ${x^2} + bx + c = 0$ and ${x^2} + qx + r = 0$ be the same, then

If $\left| {\begin{array}{*{20}{c}}{x - 4}&{2x}&{2x}\\{2x}&{x - 4}&{2x}\\{2x}&{2x}&{x - 4}\end{array}} \right| = \left( {A + Bx} \right){\left( {x - A} \right)^2},$ then the ordered pair $\left( {A,B} \right) = $. . . . .

If $a$ and $b$ are chosen randomly from set $\{1,2 ,3,4,5,6\}$ with replacement. Then probability that $\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{{a^x} + {b^x}}}{2}} \right)^{\frac{2}{x}}}=6$ is