Problem 65 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the relative maxima and relative minima, if any, of each function. $$ g(x)=x^{4}-4 x^{3}+8 $$

Step-by-Step Solution

Answer

The function \(g(x) = x^4 - 4x^3 + 8\) has a relative minimum at \(x = 3\). The Second Derivative Test is inconclusive for the critical point \(x = 0\).

Step 1: Find the derivative of the function

To find the derivative, use the power rule for differentiation. The derivative of \(g(x)\) with respect to \(x\) is: \[g'(x) = \frac{d}{dx}(x^4 - 4x^3 + 8) = 4x^3 - 12x^2\]

Step 2: Find the critical points

To find the critical points, set the derivative equal to zero and solve for \(x\). \[4x^3 - 12x^2 = 0\] Factor out \(4x^2\): \[4x^2(x - 3) = 0\] So, the critical points are \(x = 0\) and \(x = 3\).

Step 3: Find the second derivative

To determine whether the critical points are relative maxima or minima, take the second derivative of the function: \[g''(x) = \frac{d^2}{dx^2}(4x^3 - 12x^2) = 12x^2 - 24x\]

Step 4: Apply the Second Derivative Test

Evaluate the second derivative at each critical point to classify its nature: - For \(x = 0\): \[g''(0) = 12(0)^2 - 24(0) = 0\] As the second derivative at \(x = 0\) is zero, the test is inconclusive. - For \(x = 3\): \[g''(3) = 12(3)^2 - 24(3) = 36\] As the second derivative at \(x = 3\) is greater than zero, the critical point (\(x = 3\)) corresponds to a relative minimum. In conclusion, the given function \(g(x)\) has a relative minimum at \(x = 3\). The Second Derivative Test is inconclusive for the critical point \(x = 0\).

Key Concepts

Critical Points

Understanding critical points is crucial in calculus, especially when you're hunting for relative maxima and minima. Critical points are points on a graph of a function where the derivative is zero or undefined. These points can potentially represent the peaks or valleys of the function, which are maxima or minima points, respectively.

• To find critical points, you take the derivative of the function, which involves calculating the slope at any instance.
• Once you have the derivative, you set it to zero and solve for the variable. This step essentially finds where the slope is flat, indicating a potential turning point.

In our step-by-step solution, we found the critical points of the given function by setting the derivative to zero: \[4x^3 - 12x^2 = 0\]Factoring it gives us: \[4x^2(x - 3) = 0\], resulting in critical points at \(x = 0\) and \(x = 3\). Recognizing these critical points helps us determine where further analysis is needed to find relative extrema.

Second Derivative Test

The Second Derivative Test is a practical method to classify the critical points identified in a function. The test tells us whether a critical point is a relative maximum, minimum, or if the test is inconclusive.

• If the second derivative at a critical point is positive, it indicates that the function is concave up at that point, and hence, there's a relative minimum.
• If it's negative, the function is concave down, indicating a relative maximum.
• If the second derivative is zero, the test is inconclusive, meaning we may need other tests or analysis to determine the nature of the point.

In our example, evaluating the second derivative at \(x = 0\) gave a zero value, so it was inconclusive.
For \(x = 3\), the second derivative was positive, indicating a relative minimum. Using this test helped us deduce that the function \(g(x)\) has a relative minimum at this point.

Power Rule for Differentiation

The power rule is one of the simplest, yet most powerful tools in calculus for finding derivatives. It's particularly useful because it lets you quickly and efficiently differentiate terms of the form \(x^n\).

• The power rule states that the derivative of \(x^n\) is \(n \times x^{n-1}\).
• This means you multiply the exponent by the base, and then you subtract one from the exponent.
• For a term like \(x^4\), using the power rule gives \(4x^3\).

In our solution, we applied the power rule to derive \(g'(x)\), starting with: \[g(x) = x^4 - 4x^3 + 8\]to find \[g'(x) = 4x^3 - 12x^2\].Using the power rule allows you to easily handle polynomial terms, making it a cornerstone technique in calculus for analyzing polynomial functions.