Prove that $ f( x) = x^3 - 3x +c$ never has two roots
in $ [0,1] $ no matter what $c$ is.
Tags: first-diff-test, monotonicity, rolle’s-theorem, cubic, proof, t1, c3
Simple application of the derivative illustrates this. Students won’t see this right away. After they’ve thought about it for a while, I may give them a hint, “Try taking the derivative. What do you know?”
– James Epperson 1998-08-09
Although this is primarily a problem which involves the question of monotonicity and the use of the first derivative, it can also be used to remind students about transformations, in particular about the fact that adding a constant merely shifts the function up or down.
– Jeff Hildebrand 1998-08-09