Science:Math Exam Resources/Courses/MATH100/December 2018/Question 05 (a)/Solution 1

We take the derivative $f^{\prime }(x)=\frac{9}{7}(x^{\frac{2}{7}}+x^{-\frac{5}{7}})=\frac{9}{7}\cdot \frac{x+1}{x^{\frac{5}{7}}}$. The function therefore has two critical points: at $x=-1,0$. Now using a chart of signs shows that $f^{\prime }(x)$ exactly when $x\in (-1,0)$. We conclude that $f(x)\text{ is decreasing on }(-1,0)\text{ and increasing on }(-\mathrm{\infty },-1)\cup (0,+\mathrm{\infty })$.