Science:Math Exam Resources/Courses/MATH102/December 2011/Question 06

MATH102 December 2011

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Question 06
Let $f(x)=\ln(x^{3})$ and $g(x)=(\ln x)^{3}$. Find all values of x where the tangent lines of these two functions have the same slope (or show that no such values 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?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Remember that tangent line slopes are just derivatives at points. So take derivatives and set them equal.

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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. First, we take derivatives and see that $\displaystyle f'(x)={\frac {1}{x^{3}}}(3x^{2})={\frac {3}{x}}$ $\displaystyle g'(x)=3(\ln(x))^{2}{\frac {1}{x}}={\frac {3(\ln(x))^{2}}{x}}$ Setting the derivatives equal to each other gives $\displaystyle {\frac {3}{x}}={\frac {3(\ln(x))^{2}}{x}}$ which implies that $\displaystyle (\ln(x))^{2}=1$ Hence, we have that $\displaystyle \ln(x)=1$ or $\displaystyle \ln(x)=-1$. The first gives the solution $\displaystyle x=e$ and the second gives $\displaystyle x=e^{-1}$ both seen by exponentiating both sides. Both of these values are in the domain of the original function and so this completes the proof.

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