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

We can see from $f^{\prime }(x)=\frac{9}{7}\cdot \frac{x+1}{x^{\frac{5}{7}}}$ that $f^{\prime }(x)=0$ if $x=-1$ and undefined if $x=0$. We also know $f(x)$ is increasing for $x\in (-\mathrm{\infty },-1)$ and decreasing for $x\in (-1,0)$. Therefore, a local maximum occurs at $x=-1$. Moreover, a local minimum occurs at $x=0$ since $f(x)$ is decreasing for $x\in (-1,0)$ and increasing for for $x\in (0,\mathrm{\infty })$. Hence, $f(x)\text{ has a local minimum at }(0,0)\text{ and a local maximum at }(-1,\frac{7}{2})$.