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