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

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

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

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

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

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