Question

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

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

Answer 1

The correct option is C $(-\infty ,0)\cup (\frac{14}{15},\infty )$

Given:
$f\left(x\right)=(3x-7)x^{2/3}$
$\Rightarrow f\left(x\right)=3x^{5/3}-7x^{2/3}$
Differentiating w.r.t. $x$
$f^{\prime }\left(x\right)=5x^{2/3}-\frac{14}{3x^{1/3}}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{15x-14}{3x^{1/3}}>0$


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