Problem 21 - Essential Calculus

Question

In Exercises \(21-40,\) find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=x^{4}-4 x^{3}+2\)

Step-by-Step Solution

Answer

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

Step 1: Find the First Derivative

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

Step 2: Find the Critical Points

Set the first derivative equal to zero: \(4x^{3}-12x^{2}=0\). In order to solve this for \(x\), factor out the common factors from each term: \(4x^{2}(x-3)=0\). This gives two possible solutions: \(x=0\) and \(x=3\). These are the critical points of the function.

Step 3: Apply Second Derivative Test

The second derivative of the function can be obtained from the first derivative by applying the power rule once again. This results in \(f''(x)=12x^{2}-24x\). Then, substitute the critical points \(x=0\) and \(x=3\) into the second derivative. If the output is positive then the critical point is a relative minimum, while a negative output indicates a relative maximum. For \(x=0\), \(f''(x)=0\), which does not provide a conclusive result, thus the Second Derivative Test fails for this critical point. For \(x=3\), \(f''(x)=12*3^2-24*3=36>0\), indicating this critical point is a relative minimum.

Key Concepts

Second Derivative Test

The Second Derivative Test is a handy tool for analyzing the concavity of a function at its critical points and determining whether these points are relative minima or maxima.

After finding the second derivative (which, in our case, is \(f''(x) = 12x^2 - 24x\)), we check the sign of \(f''(x)\) at each critical point. A positive \(f''(x)\) implies that the graph of the function is concave up at that point, indicating a relative minimum. Conversely, if \(f''(x)\) is negative, the graph is concave down, and we have a relative maximum. Importantly, if \(f''(x)\) is zero, the test is inconclusive, and we may need to use other methods to classify the critical point, such as the First Derivative Test or analyzing the function's graph. For our function \(f(x)=x^4-4x^3+2\), applying the Second Derivative Test at \(x=3\) revealed a relative minimum since \(f''(3) = 36 > 0\). However, at \(x=0\), the test was inconclusive since \(f''(0) = 0\).

Critical Points

Critical points are where a function's first derivative is either zero or undefined, and they signify potential locations of relative extrema (minima or maxima) or points of inflection. To find a function's critical points, you first need to calculate its first derivative and then solve for where this derivative is equal to zero.

In our exercise, after calculating the first derivative \(f'(x) = 4x^3 - 12x^2\), we set it to zero and find \(x=0\) and \(x=3\) as our critical points. These are the candidates for relative extrema of the function \(f(x)\). Once identified, additional tests, like the Second Derivative Test, are used to determine the nature of these critical points.

First Derivative

The first derivative of a function is the backbone of differential calculus and gives us the rate of change of the function with respect to its input variable. In simpler terms, it describes the slope of the tangent to the function's graph at any given point and can tell us where the function is increasing or decreasing.

For example, in our function \(f(x)=x^4-4x^3+2\), we determine the first derivative (using the power rule) to be \(f'(x)=4x^3-12x^2\). By analyzing the sign of \(f'(x)\), we can conclude that when \(f'(x) > 0\), the function is increasing, and when \(f'(x) < 0\), it is decreasing. The first derivative also helps us locate the critical points where the function's behavior might change, which is crucial for finding relative extrema.

Power Rule

The power rule is a fundamental rule in calculus for differentiating functions of the form \(x^n\), where \(n\) is any real number. The rule states that the derivative of \(x^n\) is \(nx^{n-1}\). This simplifies the process of finding derivatives for polynomial functions significantly.

Applying this rule to the function \(f(x)=x^4-4x^3+2\), we differentiate each term separately to get the first derivative \(f'(x)\), which is \(4x^3-12x^2\). We also use the power rule a second time to find the second derivative \(f''(x)=12x^2-24x\), which is necessary for the Second Derivative Test. Understanding the power rule makes finding derivatives straightforward, allowing us to further investigate functions for critical points and extrema.