I came across a question earlier this day, that I did not manage to solve. I have been asked to prove that the equation $\sin(x) = ax + b$ has at least one real root, for all $a, b$, where:

1) $a$ is not zero.

2) $b$ is a real number.

I have tried using the Intermediate Value Theorem in order to solve the problem, but without much luck.

Hoping for some help here, Thanks in advance!

  • 1
    $\begingroup$ What function(s) have you tried when applying the IVT? $\endgroup$ – pjs36 Jan 31 '16 at 18:36

Hint: the function $f$ defined by $f(x) = ax + b - \sin x$ is continuous, and has limits $-\infty$ and $+\infty$ respectively at $-\infty$ and $+\infty$ (if $a > 0$; the other way if $a < 0$).

Apply the IVT on $f$.

  • $\begingroup$ Wow. I can't believe I haven't figured it out myself. Thanks a lot mister! $\endgroup$ – Chen Mor Jan 31 '16 at 18:55
  • $\begingroup$ Glad this helped! $\endgroup$ – Clement C. Jan 31 '16 at 18:55

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