Suppose that f is continuous, that f(x)=0 only for x=a,?

Question

and that f(x)>0 for some x>a as well as for some x What can be said about f(x) when x does not equal a??

Again, assume that f is continuous and that f(x)=0 only for x=a, but suppose instead that f(x)>0 for some x>a and f(x)<0 for some x

Answer 1

For your first scenario, I'd say your function has a minimum at f(a). You can also say the function is never negative.

For your second scenario, I don't think you can say anything too definite, outside of your function having only one real root. For example, it could be a wavy function that happens to cross the x axis only once. I suppose you CAN say the function is increasing at f(a). However, at f(b), where b is not equal to a, you cannot say for sure if the function is increasing or decreasing.

Answer 2

first case: f(x) has a min pt at x=a
therefore f(x)>0 when x is not equal to a
second case: f(x) has a pt of inflexion at x=a
f(x) is an increasing function