What are the absolute minimum and maximum values of the function f(x)=coshxf(x)=cosh⁡x?




Question 1Select one:

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


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


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The absolute maximum value is 1, the absolute minimum value is -1.


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The absolute maximum and minimum values do not exist.

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.

What are the absolute minimum and maximum values of the function f(x) = cosh x?

Explore the absolute minimum and maximum values of the hyperbolic cosine function f(x) = cosh x. Learn why the function has no upper bound and how it achieves a minimum value of 1 at x = 0. This explanation is suitable for high school students in Grades 11-12.

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