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To understand critical numbers, we begin by finding a function's first derivative. The first derivative reveals the rate of change of the function with respect to the variable. For example, given the function \( f(x) = x + \frac{9}{x} \), the first derivative is found using differentiation rules. We differentiate each term independently and apply the power rule and quotient rule to find \( f'(x) = 1 - \frac{9}{x^2} \). This derivative expression helps us determine where the function's slope is zero or undefined.

Finding where \( f'(x) = 0 \) or is undefined gives us the critical numbers. For \( f'(x) = 1 - \frac{9}{x^2} \), by setting \( 1 - \frac{9}{x^2} = 0 \) and solving, we find the critical numbers \( x = 3 \) and \( x = -3 \). These points are where the function's slope changes, indicating potential maxima, minima, or points of inflection.

The second derivative test is a powerful tool in calculus used to classify the critical numbers of a function. After finding the critical numbers, the next step is to determine whether these points represent a relative maximum, minimum, or neither. We achieve this by finding the second derivative of the function.

For the function \( f(x) = x + \frac{9}{x} \), the second derivative \( f''(x) = \frac{18}{x^3} \) tells us about the concavity of the function at the critical numbers. We plug the critical numbers into this second derivative:

• If \( f''(x) > 0 \) at a critical number, the function is concave up, indicating a relative minimum.
• If \( f''(x) < 0 \) at a critical number, the function is concave down, indicating a relative maximum.

This test helps us to confirm whether the critical points are indeed where the function changes from increasing to decreasing, or vice versa.

Understanding relative maximum and minimum is essential in calculus, as these represent the highest or lowest points in a function over a certain interval. In our example function \( f(x) = x + \frac{9}{x} \), we've found two critical points: \( x = 3 \) and \( x = -3 \). The second derivative test helps us identify the nature of these points.

By evaluating the second derivative at \( x = 3 \) and \( x = -3 \):

• Since \( f''(3) = \frac{2}{3} > 0 \), \( x = 3 \) is a point of relative minimum. The function is concave up at this point.
• Since \( f''(-3) = -\frac{2}{3} < 0 \), \( x = -3 \) is a point of relative maximum. The function is concave down at this point.

These designations of relative maximum or minimum mean that at \( x=3 \), the function reaches a low point, and at \( x=-3 \), it reaches a high point when compared locally to the nearby points.

In calculus, the quotient rule is a method used to differentiate functions that are the ratio of two differentiable functions. This rule is essential when dealing with fractions, as seen in our function \( f(x) = x + \frac{9}{x} \). Although our function is partially a quotient, we apply basic differentiation rules separately to each term due to its simplicity.

The quotient rule states that for a function \( g(x) = \frac{u(x)}{v(x)} \), the derivative \( g'(x) \) is given by:\[ g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \]For instance, if we had a complex quotient form, this rule would have been applied directly. However, in simpler terms like \( \frac{9}{x} \), other differentiation rules like the power rule, which transforms it to \( 9x^{-1} \), are more straightforward to use.

This rule is vital in exploring how changes in the numerator and denominator of a fraction impact the overall function.