For the equation x+sin(x)=1 does the intermediate value theorem show at least one solution on the interval [0,pi/6]?

Question

For the equation x+sin(x)=1 does the intermediate value theorem show at least one solution on the interval [0,pi/6]?

a) Yes it shows there must be at least one solution

b) No it is not conclusive

c) No it shows no solutions

Answer 1

I've already solved this problem...

define f(x) = x + sin(x)-1

f(0) = -1

f(pi/6) = pi/6 + sin(pi/6)-1

= pi/6 + 1/2 - 1

= pi/6 - 1/2

Now pi > 3, so pi/6 > 3/6 = 1/2...

so pi/6 - 1/2 > 0

So f(0) = =1 < 0 and f(pi/6)>0

so yes there is a solution by IVT