a peek at the graph might guide you along
http://www.wolframalpha.com/input/?i=plot+y+%3D++2x+%2B+1+%E2%88%92+sin%28x%29
Let f(x) = 2x + 1 − sin(x), how many roots does f(x) have in the interval [−π, π]? Use the next steps to prove that it has only one root.
a) Use the Intermediate Value Theorem to show that f(x) has at least one root.
(b) Explain why f(x) is increasing on the interval [−π,π].
(c) Use the Mean Value Theorem or Rolle’s Theorem, along with your answer from part (b), to show that f(x) cannot have two roots in the interval [−π, π]. (Hint: If there are two roots, what might go wrong?)
1 answer
1 answer