Question

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

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

Answer 1

Answer: (B)

$f(x)=(3 x-7) x^{2 / 3}$
$\Rightarrow f(x)=3 x^{5 / 3}-7 x^{2 / 3}$
$\Rightarrow f^{\prime}(x)=5 x^{2 / 3}-\frac{14}{3 x^{1 / 3}}$
$ =\frac{15 x-14}{3 x^{1 / 3}} >0$

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