The function $f(x) = \cosh x$ represents the hyperbolic cosine function, defined as:

$\cosh x = \frac{e^x + e^{-x}}{2}$

Step 1: Analyzing the Range of $f(x) = \cosh x$

1. The hyperbolic cosine function, $\cosh x$, always produces non-negative values because it’s defined as a sum of exponentials divided by 2.
2. The minimum value of $\cosh x$ occurs at $x = 0$, where $\cosh 0 = 1$.
3. As $x \to \pm \infty$, $\cosh x \to \infty$, meaning there is no maximum value for $f(x)$ on the entire real line.

Conclusion

• Absolute Minimum: 1 (at $x = 0$)
• Absolute Maximum: Does not exist (since $\cosh x \to \infty$ as $x \to \pm \infty$).

Answer:

The correct choice is:

The absolute maximum value does not exist, the absolute minimum value is 1.

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Would you like further clarification? Here are some related questions:

1. How is the range of hyperbolic functions like $\cosh x$ determined?
2. What are the main properties of hyperbolic functions?
3. How does $\cosh x$ compare to the trigonometric cosine function?
4. Why does $\cosh x$ lack an upper bound?
5. What is the behavior of $\cosh x$ as $x \to \pm \infty$?

Tip: The hyperbolic cosine function $\cosh x$ is often used to model shapes that naturally form curves, like suspension bridges and hanging cables.