Problem 18 - Essential Calculus

Question

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=x^{2}+8 x+10 $$

Step-by-Step Solution

Answer

The critical number of the function is -4. The function is decreasing on the interval \(-\infty, -4\) and increasing on the interval \(-4, \infty\). There is a relative minimum at \(x = -4, y = -6\). The graphical representation of the function confirms these results.

Step 1: Differentiate the Function

First, differentiate the function \(f(x) = x^{2} + 8x + 10\). By applying the power rule, it is obtained that \(f'(x) = 2x + 8\)

Step 2: Find the Critical Numbers

Next, determine the critical numbers by setting the derivative equal to zero and solving for \(x\). Solve \(2x + 8 = 0\), which simplifies to \(x = -4\). Therefore, the critical number is -4.

Step 3: Determine the Intervals of Increase or Decrease

Now, define the intervals based on the critical number, which are \(-\infty, -4\), and \(-4, \infty\). Pick a number from each of these intervals and substitute into the derivative. If \(f'(x)\) is positive, the function is increasing. If \(f'(x)\) is negative, the function is decreasing. For the interval \(-\infty, -4\), let \(x = -5\). Substituting, we find that \(f'(-5) = -2\) which is negative. Hence, the function is decreasing in this interval. For the interval \(-4, \infty\), let \(x = 0\). Substituting, found that \(f'(0) = 8\) which is positive. Thus, the function is increasing in this interval.

Step 4: Find the Relative Extrema

From the previous step it is known that the function transitions from decreasing to increasing at \(x = -4\), which means there is a relative minimum at \(x = -4\). Plug \(-4\) into \(f(x)\) to find that the relative minimum is \(f(-4) = -6\).

Step 5: Graph the Function

Finally, confirm these results by graphing the function using a graphing utility. The graph should show the function decreasing until \(x = -4\) and then increasing, which agrees with the relative minimum at \(x = -4, y = -6\).

Key Concepts

Increasing and Decreasing Intervals

Understanding when a function is increasing or decreasing is crucial for various applications in mathematics, from solving optimization problems to analyzing data trends. In the provided exercise, finding where the function given by the equation f(x)=x^{2}+8x+10 is increasing or decreasing involves taking its derivative and analyzing the sign of this derivative. After determining that the derivative, f'(x) = 2x + 8, has a critical point at x=-4, we test intervals around this point to decide on the behavior of the function.When we plug a number less than -4, such as -5, into the derivative, the result is negative, indicating that the function decreases as x moves towards -4. Conversely, plugging in a number greater than -4, such as 0, results in a positive derivative, meaning the function increases as x moves away from -4. This divide defines critical intervals where on (-∞, -4) the function is decreasing, and on (-4, ∞), it is increasing.

Relative Extrema

The relative extrema of a function are the points where the function reaches its local maximum or minimum values. These points are often associated with 'hills' and 'valleys' on the graph of a function. To find a relative extremum, we first locate the critical numbers where the derivative equals zero or does not exist. In our function f(x)=x^{2}+8x+10, the critical number is at x=-4, found after differentiating and setting the derivative to zero.By using the First Derivative Test, we observe that the function changes from decreasing to increasing at this critical number, defining a relative minimum. Substituting x=-4 into the original function results in the minimum value of f(-4) = -6. There are no local maxima in this particular case because the function keeps increasing as x moves towards infinity.

First Derivative Test

The First Derivative Test is a method employed to classify critical points as local extrema. To apply this test, we take the derivative of the function and analyze the change in sign around the critical points. With our function f(x)=x^{2}+8x+10, after taking the derivative and finding the single critical number at x=-4, we look at the sign of the derivative before and after this point.Before x=-4, the derivative f'(x) is negative, implying us the function is decreasing; after x=-4, the derivative is positive, suggesting the function is increasing. There's no change back to negative, so by the First Derivative Test, x=-4 is a relative minimum.

Function Graphing

Graphing a function provides a visual representation that helps to understand the behavior of the function over different intervals. With today's technology, graphing utilities can be powerful tools to confirm the analytical results obtained through calculus methods. For our function f(x)=x^{2}+8x+10, after recognizing the behavior on either side of the critical number using calculus, as described in previous sections, we can use a graphing utility to visualize these findings. The graph should show the curve descending to a lowest point—a valley—at (x, f(x)) = (-4, -6) and ascending thereafter. Confirming this visually ensures that we have accurately identified the increasing and decreasing intervals, as well as the relative minimum of our function.