Problem 31 - Single Variable Calculus

Question

Let \(f(x)=x^{4}-4 x-1\). a. Use Rolle's Theorem to show that \(f\) has exactly two distinct zeros. b. Plot the graph of \(f\) using the viewing window \([-3,3] \times[-5,5]\)

Step-by-Step Solution

Answer

a. The function \(f(x) = x^4 - 4x - 1\) is a polynomial, and thus continuous and differentiable everywhere. Its derivative, \(f'(x) = 4x^3 - 4\), has a critical point at \(x=1\). The second derivative, \(f''(x)=12x^2\), confirms a local minimum at this point. Since it is an even-degree polynomial with a positive leading coefficient, it must diverge to positive infinity at both ends, implying exactly two distinct zeros. b. Use a graphing tool to plot the function \(f(x) = x^4 - 4x - 1\) with a viewing window \([-3, 3] \times [-5, 5]\). You should observe a local minimum at x=1 and two distinct zeros where the function intersects the x-axis.

Step 1: Verify the conditions for Rolle's theorem

Rolle's theorem states that if a function is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \(f(a) = f(b)\), then there must be at least one number \(c\) in the open interval \((a, b)\) such that the derivative of the function at that point is zero i.e. \(f'(c) = 0\). The given function \(f(x) = x^4 - 4x - 1\) is a polynomial function, and all polynomial functions are continuous and differentiable everywhere.

Step 2: Compute the derivative of \(f(x)\) and solve for critical points

Let's find the derivative of the function \(f(x)\): \(f'(x) = \frac{d}{dx} (x^4 - 4x - 1) = 4x^3 - 4\) Now let's solve the equation \(f'(x) = 0\): \(4x^3 - 4 = 0\) \(x^3 = 1\) \(x = 1\) This gives us a critical point, \(x = 1\), where the function changes its concavity. We can confirm that by analyzing the second derivative. The second derivative is given by: \(f''(x) = \frac{d^2}{dx^2} (4x^3 - 4) = 12x^2\) Since \(f''(1) = 12\times 1^2 = 12 > 0\), we can conclude that there is a local minimum at \(x=1\).

Step 3: Showing exactly two distinct zeros

Now let's analyze the polynomial: 1. As we know, there is a local minimum at \(x=1\). 2. The polynomial \(f(x)\) is even degree (degree 4) and has a positive leading coefficient (the term \(x^4\)). Thus, \(f(x)\) must diverge to positive infinity as x goes to positive or negative infinity. Since we know that the function has a local minimum and both ends of the graph diverge to positive infinity, the function must intersect the x-axis. As the function passes the local minimum at x=1, it will start increasing again. Since it's an even degree polynomial, the function must cross the x-axis twice for it to go to positive infinity at both ends. Therefore, the function has exactly two distinct zeros.

Step 4: Plot the graph of \(f(x)\) for the given viewing window

To plot the graph of \(f(x) = x^4 - 4x - 1\) for the viewing window \([-3, 3] \times [-5, 5]\): 1. Choose a graphing tool (e.g., graphing calculator, online plotting tool, or software). 2. Enter the function \(f(x) = x^4 - 4x - 1\). 3. Set up the viewing window with an x-range of -3 to 3 and a y-range of -5 to 5. 4. Analyze the graph and observe the two points (zeros) where the function intersects the x-axis. The graph should show a local minimum at x=1 and two distinct zeros where the function intersects the x-axis.

Key Concepts

Polynomial Function

A polynomial function is an expression that consists of variables, coefficients, and non-negative integer exponents. It's structured as a sum of power terms, such as in the function \(f(x) = x^4 - 4x - 1\). Polynomial functions possess certain features that make them special:

• They are always continuous, meaning there are no breaks or gaps in their graph.
• They are differentiable everywhere, enabling us to find derivatives at any point.
• The degree of a polynomial, which is the highest power of the variable, determines the number of possible roots and the end behavior of the function's graph.

In the function \(f(x) = x^4 - 4x - 1\), the degree is 4, indicating it's a quartic polynomial, which generally implies up to four roots and a stable behavior at the ends of the graph (both going to infinity or negative infinity based on the leading coefficient).

Critical Points

Critical points of a function occur where the derivative is zero or undefined. For the polynomial \(f(x) = x^4 - 4x - 1\), we find its derivative as \(f'(x) = 4x^3 - 4\). Setting the derivative \(f'(x)\) to zero helps locate critical points:

• Solve \(4x^3 - 4 = 0\), which simplifies to \(x^3 = 1\). Thus, \(x = 1\) is a critical point.

At these critical points, the function may change direction or concavity. To determine the nature of \(x = 1\), look at the second derivative, which provides additional insight. This helps verify if \(x = 1\) is a local minimum, maximum, or a saddle point.

Second Derivative

The second derivative provides information about the concavity and inflection points of a function. For \(f(x) = x^4 - 4x - 1\), the second derivative is \(f''(x) = 12x^2\). This derivative shows us:

• If \(f''(x) > 0\), the function is concave up, indicating a local minimum.
• If \(f''(x) < 0\), it signifies concave down, suggesting a local maximum.

In our case, \(f''(1) = 12 \, \textgreek{mod}del; 0\), which implies a concave upward curve at \(x = 1\), meaning there is indeed a local minimum there. Understanding the second derivative and its implications helps interpret the overall shape and turning points of the polynomial graph.

Function Zeros

The zeros of a function are the x-values where the function evaluates to zero. They represent the points where the graph of the function crosses the x-axis. For \(f(x) = x^4 - 4x - 1\), using Rolle's Theorem and the behavior of polynomial curves:

• A polynomial of degree 4 like \(f(x)\) can have up to 4 zeros, but not necessarily distinct ones.
• Given the local minimum at \(x = 1\) and the end behavior (both ends going to infinity), the polynomial crosses the x-axis twice.

By graphing or using further analysis, we determine that there are precisely two distinct zeros for the polynomial function, corroborating the theorem's predictions and ensuring that the entire behavior is consistent as expected for such degree functions.