Solve the Following Questions

Question:

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

  1. $(-\infty, 0) \cup\left(\frac{3}{7}, \infty\right)$

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

  3. $\left(-\infty, \frac{14}{15}\right)$

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


Correct Option: , 2

Solution:

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

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

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

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

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

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