Problem 35 - College Algebra

Question

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. $$f(x)=x^{4}-4 x^{3}+5$$

Step-by-Step Solution

Answer

The local minimum occurs at \(x = 0\), no local maximum exists, increasing on \((0,\infty)\) and decreasing on \((-\infty, 0)\).

Step 1: Understanding the Function

The function given is \( f(x) = x^4 - 4x^3 + 5 \). It is a polynomial function of degree 4. By knowing its general shape, we can predict that it might have up to 3 turning points, forming local maxima and minima.

Step 2: Finding the Derivative

To find where the function is increasing or decreasing, we need its derivative. The derivative \( f'(x) \) of \( f(x) = x^4 - 4x^3 + 5 \) is calculated as follows: \( f'(x) = 4x^3 - 12x^2 \).

Step 3: Solving the Critical Points

Find the critical points by setting the derivative \( f'(x) = 4x^3 - 12x^2 \) equal to zero. Factor out \( 4x^2 \), obtaining \( 4x^2(x - 3) = 0 \). Solving gives \( x = 0 \) and \( x = 3 \). These are the critical points.

Step 4: Analyzing Intervals of Increase and Decrease

Use the critical points to test the intervals in \( f'(x) \). Choose a test point in each interval: \(-\infty < x < 0\), \(0 < x < 3\), and \(x > 3\). For example, f'(x) is negative in \,\(-\infty < x < 0\), \,positive in \,\(0 < x < 3\), and positive in \,\(x > 3\). The function decreases on \,\(-\infty < x < 0\), increases on \,\(0 < x < 3\) and increases on \,\(x > 3\).

Step 5: Finding Local Extrema using the Derivative Test

Apply the First Derivative Test at critical points \(x = 0\) and \(x = 3\). At \(x = 0\), the derivative changes from negative to positive, indicating a local minimum. At \(x = 3\), there is no sign change - the derivative stays positive.

Step 6: Confirming Extrema and Intervals with a Graph

Use a graphing utility to graph the function \( f(x) = x^4 - 4x^3 + 5 \). Visually check the local minimum at \(x = 0\) and the intervals where the function increases and decreases. Confirm that the graph matches the predicted behavior.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number powers. They come in the form of \[f(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\] where

• \(a_n, a_{n-1}, \, ..., a_1, a_0\) are constants
• \(n\) is a non-negative integer
• The term with the highest power, \(a_nx^n\), determines the degree of the polynomial

For example, the function \(f(x) = x^4 - 4x^3 + 5\) is a polynomial of degree 4.
This implies it could potentially have up to three turning points. The behavior near these points is crucial for identifying local extrema.

Derivative Test

The derivative test helps us understand the function's behavior between turning points.
Taking the derivative of a polynomial gives us a new function that tells us how fast and in what direction a polynomial function is changing.When working with polynomial functions like \(f(x) = x^4 - 4x^3 + 5\), the derivative is calculated term by term:

• Derive each term using the power rule: \(nx^{n-1}\)
• The derivative of \(f(x) = x^4 - 4x^3 + 5\) is \(f'(x) = 4x^3 - 12x^2\)

This tells us the rate of change of the original function. By solving \(f'(x) = 0\), we find the critical points where the function may change from increasing to decreasing, or vice versa.

Critical Points

Critical points are specific points in the domain of a function where its derivative is zero or undefined.
They indicate possible locations for local maxima, minima, or saddle points. For the function \(f(x) = x^4 - 4x^3 + 5\), after finding \(f'(x) = 4x^3 - 12x^2\), we set it to zero:

• \(4x^2(x - 3)= 0\) leads to \(x = 0\) and \(x = 3\) as critical points

These points suggest where the function's behavior might change, and they are the candidates for local extrema.

Increasing and Decreasing Intervals

To determine where the function is increasing or decreasing, we study the sign of the derivative \(f'(x)\) in the intervals divided by the critical points.
Here's how it's done step-by-step:

• Choose test points in the intervals: \((-\infty, 0)\), \((0, 3)\), and \((3, \infty)\)
• Calculate \(f'(x)\) at each test point
• For \((-\infty, 0)\), \(f'(x)\) is negative, indicating that the function is decreasing
• For \((0, 3)\), \(f'(x)\) is positive, so the function is increasing
• For \((3, \infty)\), \(f'(x)\) remains positive, and the function continues increasing

Keep in mind that these intervals help us sketch the graph and understand the function's overall behavior.