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

MATH100 December 2018

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Question 05 (c)
Let $\displaystyle f(x)=x^{\frac {9}{7}}+{\frac {9x^{\frac {2}{7}}}{2}}.$ (c) Find the intervals on which $f(x)$ is concave up and the intervals on which $f(x)$ is concave down.

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Hint
If the function is concave up, the second derivative of the function is positive. For concave down, it is negative.

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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 have $f''(x)={\frac {9}{7}}\cdot {\frac {1\cdot x^{\frac {5}{7}}-(x+1)\cdot {\frac {5}{7x^{\frac {2}{7}}}}}{x^{\frac {10}{7}}}}={\frac {9}{7}}\cdot {\frac {7x-5(x+1)}{7x^{\frac {12}{7}}}}={\frac {9(2x-5)}{49x^{\frac {12}{7}}}}.$ This denominator will always be nonnegative (because of the even number $12$ in the exponent), so it suffices to understand when the numerator is positive/negative. Note also that $f''(x)$ is undefined at $x=0$ . The numerator is zero when $x=5/2$ , positive when $x>5/2$ , and negative when $x<5/2$ . We therefore conclude that $\color {blue}f(x){\text{ is concave up on }}\left({\frac {5}{2}},+\infty \right){\text{ and concave down on }}\left(-\infty ,0\right)\cup \left(0,{\frac {5}{2}}\right)$ .