imc

Question

Consider functions f that satisfy |f(x)-f(y)| 12|x-y| for all real numbers x and y. Of all such functions that also satisfy the equation f(300) = f(900), what is the greatest possible value of f(f(800))-f(f(400))?

Accepted Answer

We have align* |f(f(800))-f(f(400))| & 12|f(800)-f(400)| &&() & 12|12|800-400|| &= 100, align* from which we eliminate answer choices (D) and (E). Note that alignat*8 |f(800)-f(300)| & 12|800-300| &&= 250, |f(800)-f(900)| & 12|800-900| &&= 50, |f(400)-f(300)| & 12|400-300| &&= 50, |f(400)-f(900)| & 12|400-900| &&= 250. alignat* Let a=f(300)=f(900). Together, it follows that align* |f(800)-a|& 50, |f(400)-a|& 50. align* We rewrite () as align* |f(f(800))-f(f(400))| & 12|f(800)-f(400)| &= 12|(f(800)-a)-(f(400)-a)| & 12|50-(-50)| &=50. align*