If f(x) is continuous in the interval [a,b] and f(a1) ,f(b1) have different signs, then the equation f(x)=0 has atleast one root between x=a1 and x=b1, where a1,b1 belongs to the interval [a,b].

since f(x) in continuous between a and b, so while x changes from a to b, f(x) must pass through all the values from f(a) to f(b) implies f(a1) to f(b1). But one of these values of f(a1) or f(b1) is positive and the other is negative , it follows that atleat for one value of x (say z) lying between a1 and b1, f(x) must be zero. Then z is the required root.