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Critical points in calculus are the values of a variable where the derivative of a function is zero or undefined. These points can indicate where a function might have a maximum or a minimum, or even a point of inflection.
To find critical points, we first need to take the derivative of the function. For example, consider the function presented in the problem:

• Initial function: \( f(x) = 10 - x - \frac{1}{x} \)
• Derivative: \( f'(x) = -1 + \frac{1}{x^2} \)

Setting \( f'(x) = 0 \) finds the critical points:

• \( -1 + \frac{1}{x^2} = 0 \)
• Solving gives \( x^2 = 1 \)
• Hence, \( x = 1 \), since only positive values are considered

The critical points are potential candidates for local maxima or minima of the function.

A derivative in calculus is a tool that measures how a function changes as its input changes. It is a fundamental concept to determine the rate at which things happen.
The derivative of a function helps identify where a function's slope is zero, which is crucial for finding critical points. In this exercise, the derivative is calculated as follows:

• Function: \( f(x) = 10 - x - \frac{1}{x} \)
• Derivative: \( f'(x) = -1 + \frac{1}{x^2} \)

The derivative \( f'(x) \) indicates how the function \( f(x) \) changes at any point \( x \). When \( f'(x) = 0 \), it tells us there might be a critical point at which function behavior can change, indicating potential maxima or minima.
To solve for this derivative and find critical points, we set \( f'(x) \) to zero and solve for \( x \). This helps determine where the function's slope is zero, guiding us to potential peaks or troughs in the function's graph.

Finding the maxima and minima of a function involves identifying the highest and lowest points the function can reach within a particular range.
In the provided exercise, by evaluating critical points and endpoints, we determine these values. After finding the critical point (\( x = 1 \)), it's essential to evaluate the function at this point and at the boundaries. Here's how:

• Calculate \( f(1) = 8 \), suggesting a possible maximum.
• Evaluate endpoints: \( f(10) = -0.1 \).
• Note behavior as \( x \) approaches 0 from the right: \( f(x) \to -\infty \).

Therefore, the maximum value of the function is 8 when \( x = 1 \), and the function approaches negative infinity as \( x \) nears zero. Understanding these points involves analyzing where the function's output reaches its highest and lowest values, guided by calculating derivatives and checking the function's limits.