Mathematics - Application of Derivative Question with Solution | TestHub

$f\left(x\right)=\left(3x-7\right)x^{\frac{2}{3}}$

For $f\left(x\right)$ is increasing $f^{\text{'}}\left(x\right)>0$.

Differentiate $f\left(x\right)$

$f^{\text{'}}\left(x\right)=3x^{\frac{2}{3}}+\left(3x-7\right)\frac{2}{3x^{{\displaystyle \frac{1}{3}}}}$

$f^{\text{'}}\left(x\right)=\frac{9x+2\left(3x-7\right)}{3x^{{\displaystyle \frac{1}{3}}}}$

$f^{\text{'}}\left(x\right)=\frac{15x-14}{3x^{{\displaystyle \frac{1}{3}}}}$

Now $f^{\text{'}}\left(x\right)>0 \Rightarrow \frac{15x-14}{3x^{{\displaystyle \frac{1}{3}}}}>0$

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