If a function $f(x)$ has a maximum at the point $(2, 4)$, what does the reciprocal of $f(x)$, which is $\frac{1}{f(x)}$, have at $x=2$?

1. Problem Description

If a function $f(x)$ has a maximum at the point $(2, 4)$, what does the reciprocal of $f(x)$, which is $\frac{1}{f(x)}$, have at $x=2$?

2. Solution Steps

If $f(x)$ has a maximum at the point $(2, 4)$, this means that $f(2) = 4$ is the largest value of $f(x)$ in some neighborhood around $x=2$. Thus, for $x$ close to $2$, we have $f(x) \le 4$.

Now consider the reciprocal function $g(x) = \frac{1}{f(x)}$.

At $x=2$, $g(2) = \frac{1}{f(2)} = \frac{1}{4}$.

Since $f(x) \le 4$ for $x$ near $2$, we have $\frac{1}{f(x)} \ge \frac{1}{4}$ for $x$ near $2$.

This means that $g(x) \ge \frac{1}{4}$ for $x$ near $2$.

Since $g(2) = \frac{1}{4}$ and $g(x) \ge \frac{1}{4}$ for $x$ near $2$, $g(x)$ has a minimum at $x=2$ with the value $\frac{1}{4}$.

Therefore, the reciprocal of $f(x)$ has a minimum at the point $(2, \frac{1}{4})$.

3. Final Answer

minimum at $(2, \frac{1}{4})$