Problem 12 - Thomas Calculus

Question

In Exercises \(9-28 :\) a. Find the intervals on which the function is increasing and decreasing. b. Then identify the function's local extreme values, if any, saying where they are taken on. c. Which, if any, of the extreme values are absolute? d. Support your findings with a graphing calculator or computer grapher. $$ h(x)=2 x^{3}-18 x $$

Step-by-Step Solution

Answer

Increasing on \((-\infty, -\sqrt{3}) \cup (\sqrt{3}, \infty)\), decreasing on \((-\sqrt{3}, \sqrt{3})\). Local max at \( x = -\sqrt{3} \), local min at \( x = \sqrt{3} \), no absolute extrema.

Step 1: Find the Derivative

To determine where the function \( h(x) = 2x^3 - 18x \) is increasing or decreasing, we first find its derivative. The derivative of the function, \( h'(x) \), gives us the slope of the tangent line at any point \( x \). Calculate \( h'(x) \):\[ h'(x) = \frac{d}{dx}(2x^3 - 18x) = 6x^2 - 18 \].

Step 2: Critical Points

Find the critical points by setting the derivative equal to zero and solving for \( x \):\[ 6x^2 - 18 = 0 \]Divide the equation by 6:\[ x^2 - 3 = 0 \]Solve for \( x \):\[ x^2 = 3 \]\[ x = \pm \sqrt{3} \]The critical points are \( x = \sqrt{3} \) and \( x = -\sqrt{3} \).

Step 3: Determine Intervals of Increase/Decrease

Use the critical points to test intervals on the number line: \((-\infty, -\sqrt{3}), (-\sqrt{3}, \sqrt{3}), (\sqrt{3}, \infty)\).Test a point in each interval in the derivative:- For \( x < -\sqrt{3} \), choose \( x = -2 \). \( h'(-2) = 6(-2)^2 - 18 = 24 - 18 = 6 \) (positive slope, function is increasing)- For \( x \in (-\sqrt{3}, \sqrt{3}) \), choose \( x = 0 \). \( h'(0) = 6(0)^2 - 18 = -18 \) (negative slope, function is decreasing)- For \( x > \sqrt{3} \), choose \( x = 2 \). \( h'(2) = 6(2)^2 - 18 = 24 - 18 = 6 \) (positive slope, function is increasing)Thus, \( h(x) \) is increasing on \((-\infty, -\sqrt{3}) \cup (\sqrt{3}, \infty)\) and decreasing on \((-\sqrt{3}, \sqrt{3})\).

Step 4: Identify Local Extreme Values

Local extreme values occur at the critical points \( x = \pm\sqrt{3} \).- At \( x = -\sqrt{3} \), the function transitions from increasing to decreasing, indicating a local maximum.- At \( x = \sqrt{3} \), the function transitions from decreasing to increasing, indicating a local minimum.Calculate the function's value at these points:\[ h(-\sqrt{3}) = 2(-\sqrt{3})^3 - 18(-\sqrt{3}) = -6\sqrt{3} + 18\sqrt{3} = 12\sqrt{3} \] (local max)\[ h(\sqrt{3}) = 2(\sqrt{3})^3 - 18(\sqrt{3}) = 6\sqrt{3} - 18\sqrt{3} = -12\sqrt{3} \] (local min)

Step 5: Identify Absolute Extremes

Since \( h(x) \) is a polynomial, it is continuous and differentiable everywhere, and tends to infinity as \( x \to \pm \infty \). Therefore, it doesn't have any absolute maximum or minimum based on the endpoints of the domain; it only has local extrema determined from the inflection of the slope.

Step 6: Graph the Function for Verification

Use a graphing calculator or software to graph \( h(x) = 2x^3 - 18x \) and verify your findings.- Observe that the graph has local extrema at \( x = -\sqrt{3} \) and \( x = \sqrt{3} \).- The function is increasing on \((-\infty, -\sqrt{3}) \cup (\sqrt{3}, \infty)\), decreasing on \((-\sqrt{3}, \sqrt{3})\), anda there is no absolute max or min.

Key Concepts

Critical Points

Critical points of a function are special points where the derivative is zero or undefined. For the function \( h(x) = 2x^3 - 18x \), we calculate its first derivative: \( h'(x) = 6x^2 - 18 \).
To find the critical points, we set the derivative equal to zero: \( 6x^2 - 18 = 0 \). Solving this equation, we get: \( x^2 = 3 \), which leads to \( x = \pm \sqrt{3} \).
These points, \( x = \sqrt{3} \) and \( x = -\sqrt{3} \), are critical points and indicate where the function's behavior changes.

Increasing and Decreasing Intervals

To determine where the function increases or decreases, we analyze the sign of its derivative \( h'(x) = 6x^2 - 18 \) around the critical points. We check intervals divided by the critical points:

• \(( -\infty, -\sqrt{3} )\): Choose a test point, like \( x = -2 \). Since \( h'(-2) = 6 \), the function is increasing here.
• \(( -\sqrt{3}, \sqrt{3} )\): Choose a test point, like \( x = 0 \). \( h'(0) = -18 \), indicating the function is decreasing in this interval.
• \(( \sqrt{3}, \infty )\): Choose a test point, like \( x = 2 \). Since \( h'(2) = 6 \), the function is increasing.

Therefore, \( h(x) \) is increasing on \((-\infty, -\sqrt{3}) \cup (\sqrt{3}, \infty)\) and decreasing on \((-\sqrt{3}, \sqrt{3})\).

Local Extrema

Local extrema on a function occur at critical points where the function changes its direction from increasing to decreasing or vice versa. For the function \( h(x) = 2x^3 - 18x \), critical points exist at \( x = -\sqrt{3} \) and \( x = \sqrt{3} \).
- At \( x = -\sqrt{3} \), the function moves from increasing to decreasing, indicating a local maximum. Calculating \( h(-\sqrt{3}) \), we find the value is \( 12\sqrt{3} \).
- At \( x = \sqrt{3} \), the function changes from decreasing to increasing, showing a local minimum. The value here is \( -12\sqrt{3} \).
These points are where local extremes occur.

Polynomial Function

Polynomial functions, such as \( h(x) = 2x^3 - 18x \), are continuous and differentiable for all real numbers. They exhibit a universal domain of \(( -\infty, \infty )\) and cover a limitless range.
The structure of polynomial functions allows for identifying behaviors like increasing/decreasing sections and local extrema. Their continuous nature means there are no jumps or breaks, even at extreme values, and their end behaviors approach infinity.

Derivative Analysis

Derivative analysis helps find critical points, increasing/decreasing intervals, and local extremes. Derivatives provide the slope of the tangent line at any point on the function \( h(x) \):\( h'(x) = 6x^2 - 18 \).
By setting \( h'(x) \) to zero, we find where the slope might change, indicating potential critical points. Evaluating \( h'(x) \) at various intervals shows where the function is increasing or decreasing. Analyzing these properties aids significantly in understanding the overall shape and characteristics of polynomial functions.