The Intermediate Value Theorem guarantees a zero between x=ax=ax=a and x=bx=bx=b if:
f(a)f(a)f(a) and f(b)f(b)f(b) are both positive
f(a)f(a)f(a) and f(b)f(b)f(b) have opposite signs
f(a)=f(b)f(a) = f(b)f(a)=f(b)
f′(c)=0f'(c) = 0f′(c)=0