Question

How do you use the Intermediate Value Theorem to show that the polynomial function `sin x + cos x - x = 0` has a real solution?

Answer 1

See the explanation.

Explanation:

That is not a polynomial function. It is also not a polynomial equation.

It is an equation that has a real solution.

The function `f(x) = sinx+cosx-x` has zeros exactly where the equation has solutions.

`f` is the sum and difference of functions that are continuous at every real number, so `f` is continuous at every real number.

Now we need a closed interval for which `f` at one end is positive and `f` at the other end is negative.

Find such an interval using the special angles (and angles coterminal with them) that you learned in trigonometry.

`0` is usually a good place to start. `f(0)=sin0+cos0-0=1`.

Now find a number where `f` is negative and apply the Intermediate Value Theorem.