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

MATH100 December 2018

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Question 05 (b)
Let $\displaystyle f(x)=x^{\frac {9}{7}}+{\frac {9x^{\frac {2}{7}}}{2}}.$ (b) Find the $x$ and $y$ for all the point(s) where $f(x)$ has a local minimum or local maximum, clearly indicating which point(s) are local minima and which point(s) are local maxima.

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Hint
What are the critical points? Remember to consider value(s) $c$ where $f'(c)$ is undefined as well.

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We can see from $f'(x)={\frac {9}{7}}\cdot {\frac {x+1}{x^{\frac {5}{7}}}}$ that $f'(x)=0$ if $x=-1$ and undefined if $x=0$ . We also know $f(x)$ is increasing for $x\in (-\infty ,-1)$ 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 $x\in (0,\infty )$ . Hence, $\color {blue}f(x){\text{ has a local minimum at }}(0,0){\text{ and a local maximum at }}\left(-1,{\frac {7}{2}}\right)$ .