Question

Show that the equation x3 + 4x = 1 has a solution between x = 0 and x = 1

Answer 1

Intermediate Value Theorem: Suppose that f(x) is an arbitrary, continuous function on an interval [a,b] . If there exists a value L between f(a) and f(b) , then there exists a corresponding value c∈(a,b) , such that f(c)=L

f(x)=x3+4x−1

f(0)=−1f(1)=4

Since the function changes sign in the interval (0,1) , hence there exists a c∈(0,1) such that f(c)=0