Question

In Exercises, find all relative extrema of the function. $$ f(x)=x^{4}-12 x^{3} $$

Short answer

The function \(f(x)=x^{4}-4x^{3}+2\) has a relative minimum at \(x = 3\). There is no relative maximum.

Step-by-step solution

Find the Derivative of the Function

The derivative of the function \(f(x)=x^{4}-4x^{3}+2\) is found using the power rule. The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). Therefore, the derivative, \( f'(x) \), is \(4x^{3}-12x^{2}\).

Set the Derivative Equal to Zero and Solve for \(x\)

Setting the derivative \( f'(x) = 4x^{3}-12x^{2} = 0 \) equal to zero and solving for \(x\) gives two solutions: \(x = 0\) and \(x = 3\). This indicates that there might be extrema at these points.

Calculate the Second Derivative of the Function

Just like in Step 1, take the derivative of the function again to get \(f''(x) = 12x^{2}-24x\).

Substitute \(x\) in Second Derivative

Substitute the values of \(x\) obtained in Step 2 into the second derivative. For \(x = 0\), \(f''(0) = 12(0)^{2}-24(0) = 0\) and is therefore neither maxima nor minima. For \(x = 3\), \(f''(3) = 12(3)^{2}-24(3) = 36 > 0\), indicating a relative minimum.

Key concepts

Second-Derivative Test

The Second-Derivative Test is a method used in calculus to determine the nature of critical points found on a graph of a function. It offers a practical way to examine whether a point is a local maximum, a local minimum, or neither. This test is often used after finding where the first derivative, or slope, equals zero.

To apply the Second-Derivative Test:

• Find the second derivative of the function, denoted as \(f''(x)\).
• Substitute the critical points into the second derivative.
• If \(f''(x) > 0\) at a critical point, the function has a local minimum there.
• If \(f''(x) < 0\) at a critical point, the function has a local maximum there.
• If \(f''(x) = 0\), the test is inconclusive, and you might need to examine further to determine the nature of the point.

This test is invaluable because it cuts down on the work needed to sketch or precisely analyze functions, especially for polynomials and rational functions in calculus problems.

Relative Extrema

In calculus, relative extrema refer to the highest or lowest points within a certain interval of a function's curve. These points are also known as relative maxima and minima. Unlike absolute extrema, which are the global highest or lowest points on an entire graph, relative extrema are local, meaning they only apply to a confined section of the graph.

To identify relative extrema, follow these steps:

• First, calculate the first derivative of the given function.
• Set the first derivative equal to zero to find the critical points where the function potentially has relative extrema.
• Use the Second-Derivative Test to classify these critical points as relative maxima or minima.

Recognizing these extrema is crucial when analyzing graphs, as they describe the function's behavior in specific regions. They can be particularly important in real-world applications where changes or breaks in patterns are informative.

Power Rule

The Power Rule is one of the most straightforward and commonly used rules in calculus. It simplifies the process of finding the derivative of a polynomial function, making it quicker and easier. This rule states that if you have a term \(x^n\), its derivative is \(nx^{n-1}\).

Here's how it works:

• Look at the exponent, \(n\), of the term.
• Multiply the entire term by \(n\).
• Subtract one from the exponent.

For example, if you apply the Power Rule to \(x^4\), you multiply by the exponent (4), making it \(4x^3\). This simple method significantly reduces the complexity when you are differentiating polynomials and is a fundamental skill in studying calculus.

Derivative

The derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus because it provides the slope of the tangent line to the function at any given point. Differentiation, the process of finding a derivative, reveals how a function changes, which is especially pivotal in optimization and motion problems.

In practical terms, the derivative \(f'(x)\) of a function \(f(x)\) can tell you a lot:

• When \(f'(x) > 0\), the function is increasing at \(x\).
• When \(f'(x) < 0\), the function is decreasing at \(x\).
• When \(f'(x) = 0\), the function might have a relative extrema at that point.

Calculating derivatives is a key step in many calculus problems, and using rules like the Power Rule can make the process simpler and more intuitive.