Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ g(x)=3 x^{4}+4 x^{3} \text { on }[-2,1] $$

Step-by-Step Solution

Answer

The absolute maximum value of the function \(g(x) = 3x^4 + 4x^3\) on the interval \([-2, 1]\) is 48 at \(x = -2\), and the absolute minimum value is -1 at \(x = -1\).

Step 1: Find the derivative of the function g(x)

To find the critical points of the function, we need to find its derivative. The function g(x) = 3x^4 + 4x^3. Apply the power rule for finding derivatives: \(\frac{d}{dx}(x^n) = nx^{(n-1)}\). $$ g'(x) = \frac{d}{dx}(3x^4 + 4x^3) = 3(4x^3) + 4(3x^2) = 12x^3 + 12x^2 $$

Step 2: Find the critical points

To find the critical points, set the derivative equal to zero and solve for x: $$ 12x^3 + 12x^2 = 0 $$ Factor out a 12x^2 from the equation: $$ 12x^2(x + 1) = 0 $$ Thus, the critical points are x = -1 and x = 0.

Step 3: Evaluate the function at the critical points and endpoints

Evaluate the function at the critical points x = -1 and x = 0, as well as at the endpoints of the interval x = -2 and x = 1: $$ \begin{array}{c|c} x & g(x) \\ \hline -2 & 3(-2)^4 + 4(-2)^3 = 48 \\ -1 & 3(-1)^4 + 4(-1)^3 = -1 \\ 0 & 3(0)^4 + 4(0)^3 = 0 \\ 1 & 3(1)^4 + 4(1)^3 = 7 \\ \end{array} $$

Step 4: Determine the absolute maximum and minimum values

By observing the table above, we can determine the absolute maximum and minimum values of the function g(x): - Absolute maximum value: g(x) = 48 at x = -2 - Absolute minimum value: g(x) = -1 at x = -1 Therefore, the absolute maximum value of g(x) on the interval [-2, 1] is 48, and the absolute minimum value of g(x) on the interval is -1.

Key Concepts

Derivative

A derivative is a fundamental concept in calculus that represents the rate of change of a function with respect to a variable. In simpler terms, think of it as the slope of the function's graph at any given point. When solving problems involving derivatives, you're often trying to determine how quickly something is changing.
To find the derivative of a function, we use specific rules, one of the most common being the "Power Rule." This rule states that if you have a function of the form \(x^n\), its derivative is \(nx^{(n-1)}\). Given the function \(g(x) = 3x^4 + 4x^3\), finding its derivative means applying the power rule to each term.

• For \(3x^4\), the derivative is \(3 imes 4x^{(4-1)} = 12x^3\).
• For \(4x^3\), the derivative is \(4 imes 3x^{(3-1)} = 12x^2\).

Thus, the derivative, \(g'(x)\), of the function \(g(x)\) is \(12x^3 + 12x^2\). This calculation helps us analyze where the function's slope changes, which is crucial for finding critical points.

Critical Points

Critical points are the values of \(x\) in the domain of a function where its first derivative is zero or undefined. These points are important as they indicate where the function's graph has a horizontal tangent line, potentially highlighting peaks, valleys, or points of inflection.
To find the critical points for our function \(g(x) = 3x^4 + 4x^3\), we set its derivative \(g'(x) = 12x^3 + 12x^2\) equal to zero.

• This equation simplifies to \(12x^2(x + 1) = 0\).
• By setting each factor to zero, we identify the critical points: \(x = -1\) and \(x = 0\).

These points will be used, along with the interval endpoints, to determine where the function achieves its maximum and minimum values on the given interval [-2, 1].

Maximum and Minimum Values

Maximum and minimum values, often called extrema, are the greatest and smallest values a function takes on a given interval. Absolute extrema include both endpoints and critical points within the interval. To find these values, evaluate the function at the critical points and at the ends of the interval.
For our function, \(g(x)\), we calculated values at critical points \(x = -1, 0\) and endpoints \(x = -2, 1\):

• \(g(-2) = 48\)
• \(g(-1) = -1\)
• \(g(0) = 0\)
• \(g(1) = 7\)

By comparing these values, the absolute maximum is \(48\) at \(x = -2\), and the absolute minimum is \(-1\) at \(x = -1\). This process ensures we fully understand the behavior of the function across its interval.

Power Rule

The power rule is an essential technique in calculus used to find the derivative of polynomial functions. It provides a straightforward method of calculation, particularly when dealing with powers of \(x\).
The rule states that for any function \(x^n\), its derivative is \(nx^{(n-1)}\). This means you multiply the term by its exponent, and then subtract one from the exponent to find the new power of \(x\).

• For example, applying the power rule to \(3x^4\) yields \(12x^3\).
• Similarly, for \(4x^3\), the derivative becomes \(12x^2\).

Using the power rule simplifies the process of differentiation, allowing us to efficiently find where changes occur in polynomial functions. By mastering this rule, you can swiftly tackle more complex calculus problems like finding maxima and minima.