# Bonus question involving derivatives?

• Mar 20th 2010, 05:33 PM
Solid8Snake
Bonus question involving derivatives?
Show that the point on the graph of y=lnx which is closest to the origin lies on the parabola y= -x^2.Draw a sketch. Also, find an approximate value of the x-coordinate of that point (with 4 decimals).

I have no clue how to approach this problem, can anyone guide me through this question?

• Mar 20th 2010, 06:14 PM
Prove It
Quote:

Originally Posted by Solid8Snake
Show that the point on the graph of y=lnx which is closest to the origin lies on the parabola y= -x^2.Draw a sketch. Also, find an approximate value of the x-coordinate of that point (with 4 decimals).

I have no clue how to approach this problem, can anyone guide me through this question?

You want to know the distance between any point on the graph of $\displaystyle y = \ln{x}$ and the origin. Call any point on the graph $\displaystyle (x, y) = (x, \ln{x})$.
$\displaystyle D = \sqrt{(x - 0)^2 + (\ln{x} - 0)^2}$
$\displaystyle = \left[x^2 + (\ln{x})^2\right]^{\frac{1}{2}}$
You want to minimise this distance, so differentiate, set it equal to $\displaystyle 0$, solve for $\displaystyle x$ and check using the second derivative test that it is in fact a minimum.