Using the results above and the intermediate value theorem we conclude that there is a solution to the equation

Question

Cos(x)=3-2x for some C €___________

and we know that f(a)<3<f(b) for

a=_____

b=_____

need help answering these

Answer 1

Consider the function f(x) = cos(x) + 2x, which is clearly continuous everywhere.

f(0)=1+0=1

f(π)=-1+2π≈5.283

So, somewhere between 0 and π, there is a number that when plugged into the f function returns a value of 3 (else how do the y-values go from 1 to 5.28?).

Note that f'(x) is always positive so this function is always increasing.

If you want to determine the value of c, such that f(c) = 3, you can use Newton's method for the function

g(x) = cos(x) + 2x - 3 to determine its root(s).

Looks like c ≈ 1.42967165 which is between 0 and π.

Here is a Desmos graph the uses Newton's Method:

desmos.com/calculator/bce6763316

This Desmos graph shows the point where cos(x) and 3 - 2x intersect.

desmos.com/calculator/fo9atqvsvd