Suppose that f is continuous in [a,b] and f'(x) exist and never be zero in (a,b). If f(a) and f(b) have opposite signs,

Question

show that the equation f(x)=0 has one and only one solution in (a,b)

Answer 1

If f is continuous in [a, b], that means that within the interval x=a and x=b, there are no breaks in the graph (no holes nor asymptotes).

 

If f'(x) exists, that means that there is a slope at every point within the interval (there is no x-value at which there is no slope).

 

If f'(x) never equals zero with in the interval [a, b], that means that there is never a point at which the slope is a horizontal line, or in other words, a critical point (a minimum or a maximum).

 

Because we know there are no minimums nor maximums within the internal [a, b], the graph must always be increasing or always be decreasing.

 

Because f(a) and f(b) have opposite signs, that means that one endpoint of the interval is positive, and the other endpoint is negative.

 

Therefore, because we know that the graph is continuously increasing or continuously decreasing, and one endpoint is positive while the other endpoint is negative, then the graph f(x) will equal 0 at only one point within the interval [a, b].