Science:Math Exam Resources/Courses/MATH102/December 2012/Question A 01
• QA 1 • QA 2 • QA 3 • QB 1 • QB 2 • QB 3 • QB 4 • QB 5 • QB 6 • QC 1 • QC 2 • QC 3(a) • QC 3(b) • QC 3(c) • QC 4(a) • QC 4(b) • QC 4(c) • QC 51 • QC 52 •
Question A 01 

The functions f(x) = x^{2} and g(x) = x^{3} are equal at x = 0 and at x = 1. Between x = 0 and x = 1, for what value of x are their graphs furthest apart?

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 

The question is asking about the distance between two graphs. Try sketching a picture of the two graphs (it doesn't have to be 100% accurate for this technique to work!), and identifying the distance between the two at a point between 0 and 1. Now find a formula for that distance. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

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. 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
Now this is simply a max/min problem (see the keyword "furthest apart" in the question): we are looking for the maximum of . First we find critical points by differentiating, setting equal to zero, and solving for x:
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). 