Question

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=\frac{8}{x^{2}+2} $$

Short answer

The function \( f(x) = \frac{8}{x^{2}+2} \) has one relative extremum at x = 0, which is a relative minimum. Second-Derivative Test was inconclusive and the result was confirmed through the First-Derivative Test.

Step-by-step solution

Find the derivative of the function

The derivative of \( f(x) = \frac{8}{x^{2}+2} \) is found using the quotient rule for derivatives. The quotient rule states that the derivative of \( \frac{f}{g} \) is \( \frac{g*f' - f*g'}{g^2} \). So, \( f'(x) = \frac{(x^{2}+2)*0 - 8*2x}{(x^{2}+2)^{2}} \). Simplified, the derivative is \( f'(x) = \frac{-8x}{(x^{2}+2)^{2}} \)

Find the critical points

Critical points occur where the derivative is zero or undefined. The derivative is undefined at no point in this case. Setting \( f'(x) = 0 \) and solving for x gives the critical points. \( -8x / (x^{2}+2)^{2} = 0 \) implies x = 0.

Compute the second derivative of the function

The second derivative is necessary for the Second Derivative Test. This requires utilizing the quotient rule again on \( f'(x) \). Therefore, \( f''(x) = \frac{(x^{2}+2)^{2}*0 - -8x*4(x)(x^{2}+2)}{(x^{2}+2)^{4}} - \frac{2*-8x*2(x)(x^{2}+2)}{(x^{2}+2)^{3}} \). Simplified, the second derivative is \( f''(x) = \frac{16x^{2}}{(x^{2}+2)^{3}} - \frac{16x^{2}}{(x^{2}+2)^{3}} = 0 \)

Apply the Second-Derivative Test

The value of the second derivative at the critical point determines whether that point is a local maximum or minimum. Here, \( f''(0) = 0 \). With a second-derivative value of 0, the Second-Derivative Test is inconclusive.

Confirm results through the First-Derivative Test

Since the Second-Derivative Test was inconclusive, the First-Derivative Test can be used to classify the critical point. Analyzing the signs of \( f'(x) \) on either side of x = 0 shows that \( f'(x) \) changes from negative to positive. This means that x = 0 is a relative minimum.

Key concepts

Second-Derivative Test

The Second-Derivative Test is a convenient method used in calculus to determine whether a given critical point of a function is a local maximum, local minimum, or a saddle point (neither a maximum nor a minimum). To apply this test, we first find the second derivative of the function. Then, we evaluate it at the critical points found by setting the first derivative to zero.

If the second derivative at a critical point is positive, the function has a relative minimum there; if it is negative, there is a relative maximum. However, if the second derivative at the critical point is zero, the test is inconclusive, and we may need to use other methods, such as the First-Derivative Test, to determine the nature of that critical point.

Applying the Second-Derivative Test

For the function given, we would take the critical point found from the first derivative and evaluate the second derivative at that point. In our exercise, after computing the second derivative, we find that it equals zero at the critical point x = 0, which means the test does not tell us whether there is a maximum or minimum at this point. Additional methods need to be applied to arrive at a conclusion.

Quotient Rule for Derivatives

The Quotient Rule is a technique for finding the derivative of a quotient of two functions. It is essential when you want to differentiate an expression of the form \( \frac{f(x)}{g(x)} \) where both \( f(x) \) and \( g(x) \) are themselves differentiable functions. According to the Quotient Rule, \( \left(\frac{f}{g}\right)' = \frac{g*f' - f*g'}{g^2} \) where \( f' \) and \( g' \) are the derivatives of \( f \) and \( g \), respectively.

This rule is particularly useful when dealing with complex fractions where the numerator and/or denominator involves variables. The rule extends from the product rule and the chain rule.

Using the Quotient Rule in Practice

In our example, to find the first derivative of \( f(x) = \frac{8}{x^{2}+2} \) we apply the Quotient Rule since the function is a ratio of two differentiable functions. This rule simplifies the process and gives us a way to handle derivatives of complex rational functions efficiently.

Critical Points in Calculus

Critical points play a significant role in the study of calculus. They are the x-values where the first derivative of a function is either zero or undefined. These points are critical in analyzing the behavior of functions since they are often locations of maximums, minimums, or inflection points.

Finding critical points is a multi-step process where you first find the derivative of the function, then solve for the values of x that make the derivative equal to zero or lead to an undefined derivative. These values are analyzed further to understand the nature of the critical points.

Identification and Analysis of Critical Points

In the example with the function \( f(x) = \frac{8}{x^{2}+2} \), we find the critical point by setting the first derivative equal to zero and solving for x. We find that x = 0 is the only critical point for this function. To determine what kind of extremum this point is, we use methods like the Second-Derivative Test or the First-Derivative Test, as simply finding the critical point is not enough to describe its behavior fully.