Problem 10 - University Calculus: Early Transcendentals

Question

Answer the following questions about the functions whose derivatives are given. a. What are the critical points of \(f ?\) b. On what open intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$f^{\prime}(x)=3-\frac{6}{\sqrt{x}}, \quad x \neq 0$$

Step-by-Step Solution

Answer

The critical point is at \(x = 4\). \(f\) is decreasing on \((0, 4)\) and increasing on \((4, \infty)\). There is a local minimum at \(x = 4\).

Step 1: Find Critical Points

A critical point occurs where the derivative is zero or undefined. Set the given derivative equal to zero and solve:\[ 3 - \frac{6}{\sqrt{x}} = 0 \]Rearrange to isolate the term with \(x\):\[ \frac{6}{\sqrt{x}} = 3 \]Multiply both sides by \(\sqrt{x}\) to clear the fraction:\[ 6 = 3\sqrt{x} \]Divide both sides by 3:\[ 2 = \sqrt{x} \]Square both sides to solve for \(x\):\[ x = 4 \]The derivative is also undefined at \(x = 0\), but since \(x eq 0\) is specified, the critical point is \(x = 4\).

Step 2: Determine Intervals of Increase/Decrease

To find where the function is increasing or decreasing, analyze the sign of the derivative \(f'(x)\) around the critical points and undefined points. Use test points in each interval formed by the critical point and undefined point:- **Interval \((0, 4)\):** Choose \(x = 1\). \[ f'(1) = 3 - \frac{6}{\sqrt{1}} = 3 - 6 = -3 \] Since \(f'(1) < 0\), the function is decreasing on \((0, 4)\).- **Interval \((4, \infty)\):** Choose \(x = 9\). \[ f'(9) = 3 - \frac{6}{\sqrt{9}} = 3 - 2 = 1 \] Since \(f'(9) > 0\), the function is increasing on \((4, \infty)\).

Step 3: Identify Local Maximum and Minimum Points

Given the behavior of the derivative, determine if there are any local maxima or minima at the critical points:- At \(x = 4\), the sign of \(f'(x)\) changes from negative to positive: - \((0,4)\): \(f(x)\) is decreasing. - \((4, \infty)\): \(f(x)\) is increasing.This implies there is a local minimum at \(x = 4\), since the function moves from decreasing to increasing.

Key Concepts

Derivative

Derivatives provide us with crucial information about the behavior of a function at a particular point. They tell us how a function is changing at any given point.
The derivative of a function, denoted as \(f'(x)\), represents the slope of the tangent line to the graph of the function at a particular point. In simpler terms, it tells us how steep the graph of the function is, either climbing up or sliding down.

• If the derivative \(f'(x)\) is positive, the function is increasing.
• If \(f'(x)\) is negative, the function is decreasing.
• A derivative of zero indicates that the function has a horizontal tangent and is neither increasing nor decreasing at that point.

Finding critical points involves determining when the derivative is zero or undefined, as these points can indicate a change in direction of the function, which is where the graph peaks or dips.

Increasing Decreasing Functions

To determine where a function is increasing or decreasing, we examine the first derivative, \(f'(x)\), and identify intervals based on its sign.

For the function described by the derivative \(f'(x) = 3 - \frac{6}{\sqrt{x}}\), we:

• Find critical points where \(f'(x)\) is zero or undefined.
• Identify intervals around these points and pick test values to assess the sign of the derivative in those intervals.
• If \(f'(x) > 0\) at a test value, the function is increasing on that interval.
• If \(f'(x) < 0\) at a test value, the function is decreasing on that interval.

In our solution, testing the intervals around the critical point \(x = 4\) revealed that the function decreases on \((0, 4)\) and increases on \((4, \infty)\). This analysis helps us understand the behavior of the function around those intervals.

Local Maximum and Minimum Values

Local maximum and minimum points are where a function reaches a peak or a trough relative to the surrounding points. These are important because they provide insights into the performance and output range of the function.

To find these points, we locate where the derivative changes sign:

• If \(f'(x)\) changes from positive to negative at a point, there is a local maximum there.
• If \(f'(x)\) changes from negative to positive, there is a local minimum there.

For the given function derivative, \(f'(x)\) transitions from negative to positive at \(x = 4\), indicating a local minimum. This change from decreasing to increasing behavior confirms a bottom at \(x = 4\), signifying that at this point, the function assumes a local minimum value.