Mathematics Interview Questions (VII)
31. For , find the real roots when . Sketch the graph when is small and then when is large, and find approximations of the real roots in both cases. When else does have 3 real roots?
Solution. When , or .
Consider . Then . Notice that, when , . Hence, if is smaller, will have three roots; if is larger, will have only one root. Since when , , will have one more root than .
If have 3 real roots, that means touches the -axis at some point, say . So and at this point, giving and . Solving them simultaneously gives .
32. Sketch , where is rounded up or down in the usual way. Then sketch .
Solution. When , . We can then filp through that partial graph horizontally due to the properties of round function. can be created in a similar fashion. You can view here for a sample.
33. Is the relationship " is an integer" transitive? What about ?
Solution. The first relationship is; if 2 divides both and , then it divides , hence . The second isn't; consider, for instance, .
34. Differentiate .
Solution. Simplifying gives . Using quotient rule, .
P.S. Does the simplified form remind you of the Fibonacci sequence? Try to find a general pattern and prove it!
35. Sketch graph of .
Solution. The first two are trivial. For the third one, notice that
it crosses the -axis at (0,1),
the -axis is its asymptote,
its derivative is , negative when and positive when ,
and its sketching is also easy. You can view here for a sample.