Question

Consider the equation x^5-cos(pi x)=0 . We can use the Intermediate Value Theorem to prove that the given equation has a root between 0 and 1. Answer the following questions.
1.show equation can’t have two real roots between 0 and 1.
2.Use Newton's method with the initial approximation x1=1/2 to find the second approximation x2 to the root of the given equation.

Answer 1

we know that
f(0) = -1
f(1) = 2
so there's a zero
f'(x) = x^4 + πsin(πx) is always positive in (0,π) so f(x) is strictly increasing

Now just apply Newton's Method to f(x).