Minimum value of a continuous function.

Question

Let $f:[a, b]\to (0, \infty)$ be a continuous function. Let $$F:[a, b]\times [a, b]\to (0, \infty): F(x, y) =\frac{f(x)}{f(y)}$$ Then I am interested in the lower bound on $F$. If it is $1$ then how to show it?

Answer 1

I think we can have a better lower bound that $0$. Since $\left[ a, b \right]$ is compact and $f$ is continuous, $f$ attains it maximum and minimum value in some interval. Let us name it $M$ and $m$ respectively. Then,

$$m \leq f \left( x \right) \leq M$$

This gives $F \left( x, y \right) \geq \dfrac{m}{M}$

Hence, $\dfrac{m}{M}$ will be your required lower bound.