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The given function is:

\[
h(x, y) = 1 - e^{-\left(x^2 + y^2 - 5x\right)}
\]

To approximate the coordinates of the peak or depression, you need to analyze the function's behavior. Since the problem mentions "one isolated peak or one isolated depression," it suggests the presence of a local maximum or minimum.

This function has a term that involves an exponential, \( e^{-(x^2 + y^2 - 5x)} \), which decreases rapidly for large values of \( x \) or \( y \). The isolated peak or depression likely occurs near where the expression inside the exponent is minimized.

### Steps:
1. Take the partial derivatives of \( h(x, y) \) with respect to \( x \) and \( y \) and set them to zero to find the critical points.
2. Use a graphing utility to plot the function \( h(x, y) \) and inspect the coordinates of the maximum or minimum.
3. Round the coordinates to one decimal place as required.

Would you like a detailed solution for finding the critical points, or would you prefer a graphing-based approach to get the approximation?

Here are some related questions to explore this concept:
1. What are the steps for finding critical points using partial derivatives?
2. How do we classify a critical point as a maximum, minimum, or saddle point?
3. Why does the exponential function influence the overall shape of the function?
4. How does the term \( 5x \) in the exponent affect the location of the peak or depression?
5. Can this function have multiple critical points or just one?

**Tip:** Always examine the second derivatives (Hessian matrix) when trying to classify critical points!

The following function has exactly one isolated peak or one isolated depression (one local maximum or minimum). Use a graphing utility to approximate the coordinates of the peak or depression. h(x, y) = 1 - e^(-(x^2 + y^2 - 5x)). Approximate the coordinates of the peak or depression.

Explore how to find the local maximum or minimum of the function h(x, y) = 1 - e^(-(x^2 + y^2 - 5x)). This problem involves partial derivatives, critical points, and analyzing exponential functions using multivariable calculus techniques. Ideal for college students learning about extremas in functions of two variables.

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