Show that sin x + 2x has exactly one solution.

Calc! How do I do this? :/

Thank you! :)

Question

Show that sin x + 2x has exactly one solution.

Calc! How do I do this? :/

Thank you! :)

linyu · Answer

did you mean sins + 2x = 0 ?

anonymous · Answer

no... sin x  + 2x

linyu · Answer

but that is not an equation, so it stays the same

linyu · Answer

wait i made a typo i mean sinx+2x=0

anonymous · Answer

i think that's what it means haha

it just says show that sin x + 2x has exactly one solution.... :/

anonymous · Answer

because its linear? without it = to something who knows. you could assume its = to 0

anonymous · Answer

ohh okay thank you!

ganeshie8 · Answer

By inspection  $x = 0$ is one solution

Next, notice that the first derivative $\cos x + 2$ is always positive $\implies $ the given function is strictly increasing $\implies $ it cuts the x axis only once. QED.

anonymous · Answer

@ganeshie8 ohh okay...

how do you show that it has exactly one solution (x=0) though?

linyu · Answer

sinx + 2x = 0
sinx = -2x
sin(0)=0
2(0)=0

anonymous · Answer

ohh so that's all that goes into it? using calc? :O

ganeshie8 · Answer

it cuts x-axis only once  =>  it has exactly one and only one solution :)

anonymous · Answer

oh, how would i be able to show the work for that? :/ would i use a graph or something? :O