Question:
The function, $f(x)=(3 x-7) x^{2 / 3}, x \in R$, is increasing for all x lying in :
Correct Option: , 2
Solution:
$f(x)=(3 x-7) x^{2 / 3}$
$\Rightarrow \mathrm{f}(\mathrm{x})=3 \mathrm{x}^{5 / 3}-7 \mathrm{x}^{2 / 3}$
$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=5 \mathrm{x}^{2 / 3}-\frac{14}{3 \mathrm{x}^{1 / 3}}$
$=\frac{15 x-14}{3 x^{1 / 3}}>0$
$\therefore \quad \mathrm{f}^{\prime}(\mathrm{x})>0 \forall \mathrm{x} \in(-\infty, 0) \cup\left(\frac{14}{15}, \infty\right)$