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Understanding intervals of increase and decrease helps us analyze a function's behavior. To find these intervals, we use the first derivative, which tells us where the function is increasing or decreasing. Let's look at the given function: \[ f(x) = \frac{1}{x^3} + \frac{2}{x^2} + \frac{1}{x} \]The first derivative is: \[ f'(x) = -\frac{3}{x^4} - \frac{4}{x^3} - \frac{1}{x^2} \] To find where the function increases or decreases, we set the first derivative equal to zero and solve for critical points. However, in this example, there are no real solutions. Therefore, we need to look at the sign of the derivative in different intervals. We find:

• For \( x > 0 \), \( f'(x) < 0 \), so the function is decreasing.
• For \( x < 0 \), \( f'(x) < 0 \), so the function is also decreasing.

Both sides of the graph are decreasing, showing no intervals of increase in this case.

Intervals of concavity reveal where the function curves upward or downward. We use the second derivative for this analysis. The function's second derivative is:\[ f''(x) = \frac{12}{x^5} + \frac{12}{x^4} + \frac{2}{x^3} \] Just as with the first derivative, we set the second derivative to zero to find inflection points. Here, we find no real solutions either. So then, we analyze the sign of the second derivative:

• For \( x > 0 \), \( f''(x) > 0 \), the function is concave up.
• For \( x < 0 \), \( f''(x) < 0 \), the function is concave down.

Thus, the function changes concavity around the vertical asymptote at \( x = 0 \), curving upwards for positive \( x \) and downwards for negative \( x \).

Calculus derivatives are the backbone of analyzing functions. The first derivative \( f'(x) \) gives us the slope of the function, indicating where it increases or decreases. For example, \[ f'(x) = -\frac{3}{x^4} - \frac{4}{x^3} - \frac{1}{x^2} \] shows us that the slope is always negative, hence the function is always decreasing.The second derivative \( f''(x) \) shows the rate at which the slope changes. This reveals how the function curves, known as concavity. Here, \[ f''(x) = \frac{12}{x^5} + \frac{12}{x^4} + \frac{2}{x^3} \] indicates concavity. For \( x > 0 \), the positive sign means the curve is upwards (concave up), and for \( x < 0 \), the negative sign means the curve is downwards (concave down).Understanding both derivatives aids in sketching the graph accurately. This helps us capture all key features like intercepts, asymptotes, high and low points, and inflection points.