Thread: Prove there is a number that exists

Hey, I have a problem that is listed below and I know I might be rushing it a bit, simply because it hasn't been really taught yet (or to my knowledge), but the problem is below:




I know that for C) x has to be equal to zero. For A), I have no idea how to isolate for x, and for B) I think sin(x) - 1 = x. However, I do not know if this is a valid claim because of two x variables in B.

First question is how do I prove such claims? Second question is how do I even isolate for X in the first part.

Thank you.

Try applying the intermediate value theorem for part A). Depending on how much you know about continuity, you should be able to justify why the left hand side is a continuous function. Then try 2 obvious x values to get the left smaller than 198 and larger than 198.

for b) you can rearrange to get sin(x)-x and use the intermediate value theorem again

for b)
use the Taylor formula
such that:


and
f(x) = sin(x)
P(x) = x+1

you get that for x=0 (a=0) the function f(x) and the polynomial P(x) are equivalent.