Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point. $$ f(x)=2 x^{4}+64 x $$

Step-by-Step Solution

Answer

The function \(f(x) = 2x^4 + 64x\) has one critical point at \(x = -2\), which is a local minimum.

Step 1: Find the first derivative and critical points

The first derivative of the function \(f(x) = 2x^4 + 64x\) can be obtained by applying the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\). So, \(f'(x) = 8x^3 + 64\). The critical points are the x-values that make the derivative of the function equal zero or undefined. In this case, the derivative is never undefined, so we can set the derivative equal to zero and solve for x: \n\n\(8x^3 + 64 = 0\)\n\(x^3 = -8\)\n\(x = -2\). So, -2 is the only critical point.

Step 2: Find the second derivative

To find the second derivative of the function, we take the derivative of \(f'(x) = 8x^3 + 64\). Using the power rule again, we find that \(f''(x) = 24x^2\). This function is never undefined, so there's no need to check for division by zero or natural logarithm of a negative number.

Step 3: Use the second derivative test

The second derivative test states that if \(f''(x)\) is positive at a critical point, then that point is a local minimum, and if \(f''(x)\) is negative at a critical point, then that point is a local maximum. So, let's substitute our critical point -2 into the second derivative: \(f''(-2) = 24*(-2)^2 = 96\). Since this is a positive number, -2 is a local minimum according to the second derivative test.

Step 4: Use an alternative method (if necessary)

In this case, the second derivative test didn't fail, so an alternative method is not needed. If the second derivative test were inconclusive (for instance, if \(f''(x)\) were zero or undefined at a critical point), an alternative method such as the first derivative test could be used. This involves picking test points in each interval determined by the critical points and evaluating the first derivative at these points to see where the function is increasing or decreasing.

Key Concepts

Second Derivative Test

When studying calculus, the Second Derivative Test is a handy tool to determine whether a given critical point of a function is a local maximum, a local minimum, or neither. After finding a critical point of a function, say by setting its first derivative equal to zero, we can use its second derivative to provide information about the behavior of the function around that point.

Here’s how it works:

• If the second derivative at a critical point is positive (\(f''(c) > 0\)), the function is concave up at that point, indicating it's a local minimum.
• Conversely, if the second derivative is negative (\(f''(c) < 0\)), the function is concave down, suggesting a local maximum.
• If the second derivative is zero (\(f''(c) = 0\)), the test is inconclusive, and alternate methods like the First Derivative Test should be considered.

Using this approach, we evaluated that at the critical point \(x = -2\), the second derivative \(f''(-2) = 96\), a positive number, thus indicating \(-2\) as a local minimum.

First Derivative

The First Derivative of a function can often give us crucial insights into its behavior. By calculating the derivative, we can determine where the function has slopes of zero, meaning these are points where the function might be at a local maximum or minimum.

When you find the derivative of a function, such as \(f(x) = 2x^4 + 64x\), you use rules like the Power Rule. From this, we found the first derivative: \(f'(x) = 8x^3 + 64\). These points, where the derivative is zero, are known as critical points.

Once the first derivative is set to zero, equations can be solved for \(x\) to locate critical points. In our case, setting \(8x^3 + 64 = 0\) yielded \(x = -2\) as the only critical point. This method shows us where the function stops increasing or decreasing, offering a preliminary label on potential maxima or minima.

Power Rule

The Power Rule is a fundamental tool in calculus used for finding derivatives, specifically the derivative of a term in the form of \(x^n\). The rule itself states: if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). This mathematical operation is essential for simplifying the derivative-taking process.

For the function \(f(x) = 2x^4 + 64x\), the Power Rule was applied to each term:

• Derivative of \(2x^4\) becomes \(8x^3\) (where \(n=4\) and it is multiplied by 2).
• Derivative of \(64x\) becomes 64 (where \(n=1\)).

This results in the first derivative \(f'(x) = 8x^3 + 64\).

Understanding the Power Rule enables the quick assessment of polynomial derivatives, allowing for efficient analysis of function behavior without getting tangled in complex calculations.

Local Maxima and Minima

In calculus, Local Maxima and Minima are points in the domain of a function where the function reaches its highest or lowest value, respectively, within a small surrounding neighborhood. Identifying these points is vital for understanding the general shape and direction of a function.

To determine these crucial points, we use critical points found by setting the derivative to zero. After finding these critical points, tests such as the Second Derivative Test are employed. For example, in our function, \(x = -2\) was determined as a critical point where the second derivative was evaluated.

In this particular exercise, substituting \(x = -2\) into the second derivative, \(f''(-2) = 96\) shows it's positive, confirming \(-2\) is not merely a point of inflection but a local minimum. Similarly, if new critical points are determined in different problems, the same process helps classify them appropriately.

Looking for Local Maxima and Minima enables a deeper comprehension of a function's performance across its domain, highlighting prominent peaks and troughs in its graphical representation.