Use the intermediate value theorem to show that there is a r

Solution

The intermediate value theorem states the following: If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c [a, b] such that f(c) = u.

We can say f(x) = sin (x) - x² + x is continuous everywhere
f(1) = sin(1) = 0.01745
f(2) = sin(2) - 2 = -1.9651
Hence we have here by IVT: There must exist a number c between x = 1 and x = 2 for which the value of f(x) = 0

Hence proved by IVT that there is a root of the equation in (1,2)