Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ G(x)=2 x^{3}-9 x^{2}+12 x $$

Step-by-Step Solution

Answer

The function is increasing on \((- fty, 1)\) and \((2, fty)\), and decreasing on \((1, 2)\).

Step 1: Find the Derivative

To find where the function is increasing or decreasing, we first need to find its derivative. The derivative of the function \(G(x) = 2x^3 - 9x^2 + 12x\) is \(G'(x)\). Using basic differentiation rules, we find:\[ G'(x) = 6x^2 - 18x + 12 \]

Step 2: Find Critical Points

To find the critical points, we set the derivative equal to zero and solve for \(x\):\[ 6x^2 - 18x + 12 = 0 \]Dividing the entire equation by 6 simplifies it to:\[ x^2 - 3x + 2 = 0 \]This quadratic equation can be factored as:\[ (x - 1)(x - 2) = 0 \]Thus, the critical points are \(x = 1\) and \(x = 2\).

Step 3: Determine Sign of Derivative in Intervals

To determine where the function is increasing or decreasing, evaluate \(G'(x)\) in each interval determined by the critical points: \((-fty, 1)\), \((1, 2)\), and \((2, fty)\).- For \(x < 1\), choose \(x = 0\): \(G'(0) = 6(0)^2 - 18(0) + 12 = 12\), which is positive.- For \(1 < x < 2\), choose \(x = 1.5\): \(G'(1.5) = 6(1.5)^2 - 18(1.5) + 12\) results in a negative value, indicating the function is decreasing.- For \(x > 2\), choose \(x = 3\): \(G'(3) = 6(3)^2 - 18(3) + 12 = 18\), which is positive.

Step 4: Conclude Increasing and Decreasing Intervals

From the above evaluations:- The function \(G(x)\) is increasing on the intervals \((-fty, 1)\) and \((2, fty)\).- The function \(G(x)\) is decreasing on the interval \((1, 2)\).

Key Concepts

Critical Points

Critical points of a function are the values of the variable, typically denoted as \( x \), where the derivative of the function equals zero or is undefined. These points are significant since they represent places where the function may change its direction—either from increasing to decreasing or vice versa. To find critical points, follow these steps:

• Calculate the derivative of the function.
• Set the derivative equal to zero and solve for \( x \).
• Check where the derivative might be undefined, although for polynomial functions, this usually doesn't happen.

The critical points we obtain are potential candidates for local maxima, minima, or points of inflection. In the example of \( G(x) = 2x^3 - 9x^2 + 12x \), we found the derivative to be zero when \( x = 1 \) and \( x = 2 \), indicating these are the critical points. Checking these points is crucial in analyzing the function's behavior.

Increasing and Decreasing Intervals

Once the critical points have been identified, they can be used to determine intervals where the function is increasing or decreasing. The Monotonicity Theorem assists us here. This theorem states:

• If the derivative of the function, \( f'(x) \), is positive over an interval, then the function \( f(x) \) is increasing on that interval.
• If \( f'(x) \) is negative over an interval, then \( f(x) \) is decreasing on that interval.

You can find these intervals by evaluating the derivative at test points within the divided intervals set by the critical points. For example, with \( G'(x) = 6x^2 - 18x + 12 \), we saw the function decreasing between \( x = 1 \) and \( x = 2 \) and increasing outside these interval borders, specifically on \((-\infty, 1)\) and \((2, \infty)\). These insights help to visualize the graph and understand the function's overall behavior.

Derivative of a Function

The derivative of a function is a foundational concept in calculus. It provides information about the rate at which the function's value changes in response to changes in the variable. Essentially, the derivative tells us the slope of the tangent line to the function at any given point. In more practical terms, it indicates whether the function is climbing up (increasing) or going down (decreasing).

For the function \( G(x) = 2x^3 - 9x^2 + 12x \), taking its derivative means applying the rules of differentiation. The power rule, one of the simplest techniques, was applied to each term to get \( G'(x) = 6x^2 - 18x + 12 \). The derivative's sign, either positive or negative, determines the function's increasing or decreasing nature over different intervals as explored in previous steps.

Quadratic Equation

A quadratic equation is one of the simplest polynomials involving a squared term. It generally follows the form \( ax^2 + bx + c = 0 \). Solving a quadratic equation can be done using various methods, such as factoring, completing the square, or applying the quadratic formula.

In the step-by-step solution provided, the quadratic equation \( x^2 - 3x + 2 = 0 \) was factored to obtain \( (x - 1)(x - 2) = 0 \), leading to solutions \( x = 1 \) and \( x = 2 \). These solutions correspond to the critical points of the derivative function mentioned earlier. Understanding how to solve quadratic equations is crucial, as they frequently emerge in calculus and algebra, becoming a key step in finding where functions change their increasing or decreasing nature.