The function,

Question:

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

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

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

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

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


Correct Option: 1,

Solution:

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

$f^{\prime}(x)=3 x^{2 / 3}+(3 x-7) \cdot \frac{2}{3} x^{-1 / 3}$

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

For increasing function

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