Science:Math Exam Resources/Courses/MATH102/December 2012/Question A 01/Solution 1

As stated in the hint, sketching a picture of the two graphs between 0 and 1 is a good first step.

Identify the distance between the two graphs and write a formula for it.

The distance between the two graphs at a point x can be given by

$D(x)=x^{2}-x^{3}$

Now this is simply a max/min problem (see the keyword "furthest apart" in the question): we are looking for the maximum of $D(x)$.

First we find critical points by differentiating, setting equal to zero, and solving for x:

$D'(x)=2x-3x^{2}$

$0=x(2-3x)$

So we have critical points x = 0 and x = 2/3. We know x = 0 isn't the correct answer as we already know that the graphs are equal there. Thus, the point of furthest distance must be x = 2/3 or (c).