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

MATH102 December 2011

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[hide] 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).

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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!

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

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[show] Solution
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.