Question

A function \(f(x)\) is given. Find the critical points of \(f\) and use the Second Derivative Test, when possible, to determine the relative extrema. \(f(x)=x^{4}-4 x^{3}+6 x^{2}-4 x+1\)

Short answer

The critical point is \( x = 1 \), but it is a point of inflection, not a local extremum.

Step-by-step solution

Find the First Derivative

To find the critical points, start by finding the first derivative of the function. Given the function \( f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \), the first derivative is:\[ f'(x) = 4x^3 - 12x^2 + 12x - 4. \]

Set the First Derivative to Zero

To find the critical points, set the first derivative equal to zero:\[ 4x^3 - 12x^2 + 12x - 4 = 0. \]This is a cubic equation that needs to be solved to find the values of \(x\) that are critical points.

Solve the Cubic Equation

One approach to solving the cubic equation \( 4x^3 - 12x^2 + 12x - 4 = 0 \) is to use the Rational Root Theorem or synthetic division to test possible rational roots. Testing \( x = 1 \) might reveal a root, as it simplifies the equation to zero. Proceed by factoring or other methods to solve:\[ 4x^3 - 12x^2 + 12x - 4 = 4(x-1)^3. \]The root found is \( x = 1 \), which has multiplicity three.

Find the Second Derivative

Calculate the second derivative of the function to apply the Second Derivative Test. The second derivative of \( f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \) is:\[ f''(x) = 12x^2 - 24x + 12. \]

Use the Second Derivative Test

To use the Second Derivative Test, plug in the critical point into the second derivative:\[ f''(1) = 12(1)^2 - 24(1) + 12 = 0. \]Since \( f''(1) = 0 \), the Second Derivative Test is inconclusive at \(x = 1\).

Analyze Further for Extrema

Because the Second Derivative Test is inconclusive, analyze the behavior around \( x = 1 \). Since \( (x-1)^3 \) changes sign around \( x = 1 \), \( x = 1 \) is a point of inflection rather than a local minimum or maximum.

Key concepts

First Derivative

The first derivative of a function is key to identifying critical points. It represents the rate at which the function's output values change with respect to changes in input values. Critical points occur where the first derivative equals zero or does not exist, indicating potential locations for relative extrema or inflection points.
For the function given, \( f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \), the first derivative is calculated as \( f'(x) = 4x^3 - 12x^2 + 12x - 4 \).
To find critical points, solve \( 4x^3 - 12x^2 + 12x - 4 = 0 \). Solving this equation will give you the values of \( x \) where the slope of the tangent to the function is zero. This indicates potential turning points or stationary points of the graph.

Second Derivative

The second derivative provides insight into the concavity of a function and helps determine the nature of critical points. If the second derivative is positive at a critical point, it suggests that the function is concave up, indicating a local minimum. Conversely, if the second derivative is negative, it means the function is concave down, indicating a local maximum.
For the function \( f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \), the second derivative is \( f''(x) = 12x^2 - 24x + 12 \).
In our example, when we substitute the critical point \( x = 1 \) into the second derivative, \( f''(1) = 0 \), indicating that the Second Derivative Test is inconclusive. This means more investigation is needed to understand the behavior of the function at this point.

Relative Extrema

Relative extrema are the local peaks and troughs in the function graph, where the function changes from increasing to decreasing or vice versa. To find relative extrema, you use first and second derivatives:

• Identify critical points using the first derivative.
• Use the second derivative to test the nature of these points when possible.

In the example provided, the critical point \( x = 1 \) required further analysis since the Second Derivative Test was inconclusive, as \( f''(1) = 0 \). By examining the surrounding values or the expression form, you can conclude despite the test's outcome; in this case, that \( x = 1 \) is a point of inflection—not a relative extremum—because the sign of \((x-1)^3\) changes around it.

Rational Root Theorem

The Rational Root Theorem is a useful tool for finding rational roots of polynomial equations. According to this theorem, any rational root of a polynomial, in the form \( \frac{p}{q} \), must have a numerator \( p \) that is a factor of the constant term and a denominator \( q \) that is a factor of the leading coefficient.
Applying this theorem to solve the equation \( 4x^3 - 12x^2 + 12x - 4 = 0 \) helps find possible logical values for \( x \) that make the polynomial zero. Testing these values can simplify the process of factoring the polynomial or confirm real roots.
In this case, \( x = 1 \) was tested and found to be a root by simplifying the polynomial to \( 4(x-1)^3 \), where \( x = 1 \) has a multiplicity of three. This methodology efficiently assists in locating potential rational solutions before dealing with possibly complex factorization steps.