### Home > CALC > Chapter 4 > Lesson 4.5.1 > Problem 4-162

4-162.

Sketch

*f*(*x*) =*x*+ cos*x*. Homework Help ✎Use the Intermediate Value Theorem to show there is a root between −1 and 0.

Let

*x*_{1}= −1*.*Use Newton's Method to find*x*_{3}.How close is

*x*_{3}to the actual root of*f*(*x*)? Calculate the error.

State the conditions of the Intermediate Value Theorem: 'Since *f*(*x*) is continuous on the closed domain [−1, 0]...

Prove the thesis of the IVT: ... and *f*(−1) < 0 and *f*(0) > 0, there must be a value *c*, on −1 < *c* < 0, such that *f*(*c*) = 0.

Check your work: *x*_{3} = −0.739

*x*_{3} and the actual root should be very close.