Question

For the function $f(x) = x + \frac{1}{x}$ ($x \neq 0$), which of the following statements is true?

• A. local maximum value is 2
• B. local minimum value is $-2$
• C. local maximum value is $-2$
• D. local minimum value < local maximum value

Hint:

For $x > 0$, by the AM-GM inequality: $\frac{x + \frac{1}{x}}{2} \geq \sqrt{x \cdot \frac{1}{x}} = 1 \Rightarrow x + \frac{1}{x} \geq 2$. Thus, the minimum positive value is 2. For negative values, replacing $x$ with $-x$ flips the signs, making the maximum negative value $-2$.

Answer 1

The Correct Option is C