Science:Math Exam Resources/Courses/MATH100/December 2014/Question 08 (f)

MATH100 December 2014

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Question 08 (f)
Let $f(x)={\frac {\log(x)}{\sqrt {x}}}$. Note $\log(x)$ is the natural logarithmic function, also denoted $\ln(x)$ (f) The second derivative of $f(x)$ is $f''(x)={\frac {3\log(x)-8}{4x^{5/2}}}.$ Determine the intervals on which $f$ is concave up and the intervals on which $f$ is concave down. Find the coordinates of any inflection points if they exist.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

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Hint
What conditions on the second derivative will solve this?

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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. The second derivative is defined everywhere on the domain. The only zero occurs at ${\begin{aligned}3\log(x)&=8\\x&=e^{8/3}\end{aligned}}$ So we examine: ${\begin{aligned}(0,e^{8/3}):f''(x)&={\frac {neg}{pos}}=neg&{\text{concave down}}\\(e^{8/3},\infty ):f''(x)&={\frac {pos}{pos}}=pos&{\text{concave up}}\end{aligned}}$ So, the only inflection point is at $(e^{\frac {8}{3}},f(e^{\frac {8}{3}}))=(e^{\frac {8}{3}},{\frac {8}{3e^{4/3}}})$.

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