Problem 24 - College Algebra and Calculus: An Applied Approach

Question

In Exercise, use a graphing utility to estimate graphically all relative extrema of the function. $$ f(x)=-\frac{1}{3} x^{5}-\frac{1}{2} x^{4}+x $$

Step-by-Step Solution

Answer

The relative extrema of the function can be found by plotting the function, calculating its derivative and identifying the points where the derivative is zero. These points, completed with a visual analysis, provide an estimation of the extrema. Remember to analyze the curve correctly and identify whether each critical point is a maximum or minimum.

Step 1: Understand the Function

The function given is \( f(x) = -\frac{1}{3}x^{5} - \frac{1}{2}x^{4} + x \). It's a polynomial of degree 5 and it's continuous and differentiable everywhere. Thus, relative extrema can only occur at critical points of the function, where its derivative equals zero or is undefined.

Step 2: Calculate the Derivative

Differentiate the function to find the critical points. The derivative of \( f(x) = -\frac{1}{3}x^{5} - \frac{1}{2}x^{4} + x \) is \( f'(x) = -\frac{5}{3}x^{4} - 2x^{3} + 1 \). Set this derivative equal to zero and solve for x to find the x-coordinates of the critical points.

Step 3: Plot the Function

Next, plot the function using a graphing utility. This will provide a visual representation of the function, making it easier to estimate the points of relative extrema.

Step 4: Identify the Extrema

Identify the points on the graph where the graph reaches a local maximum or minimum. These are the estimated points of relative extrema. Remember that a relative minimum point is where the graph has a downward point and a relative maximum point is where the graph has an upward point.

Key Concepts

Degree of a Polynomial

In mathematics, the degree of a polynomial is one of the most important characteristics to understand. It is defined as the highest power of the variable in the polynomial expression.
For the function given in the exercise, \[ f(x) = -\frac{1}{3}x^{5} - \frac{1}{2}x^{4} + x \]the highest exponent of the variable \( x \) is 5, thus making it a fifth-degree polynomial.
The degree of a polynomial helps in determining the general shape and behavior of its graph. For instance:

• A polynomial of degree \( n \) can have up to \( n \) roots or x-intercepts.
• It can also have up to \( n-1 \) turning points, which are potential relative maxima or minima.

The degree gives us a clue about the complexity and symmetry of the curve, guiding us in analyzing its behavior more effectively.

Critical Points

Critical points are crucial for identifying where a function may have local maxima or minima, known as relative extrema. A critical point occurs where the derivative of a function is zero or undefined.
For the function \[ f(x) = -\frac{1}{3}x^{5} - \frac{1}{2}x^{4} + x \]the derivative is\[ f'(x) = -\frac{5}{3}x^{4} - 2x^{3} + 1 \]By setting \( f'(x) = 0 \) and solving for \( x \), we find the possible x-coordinates where the critical points may exist.
Critical points help determine the behavior of the graph:

• If the derivative changes signs around these points, a relative extremum occurs.
• Some critical points may not lead to any extremum, known as inflection points, where the graph merely changes its concavity.

Understanding critical points ensures that the analysis of the graph is based on solid mathematical ground.

Derivative

Derivatives serve as the backbone of finding critical points and analyzing the slope of a function. Taking the derivative of a polynomial function gives us another polynomial that indicates the rate of change of the original function.
For our polynomial function \[ f(x) = -\frac{1}{3}x^{5} - \frac{1}{2}x^{4} + x \]the derivative is simplified to \[ f'(x) = -\frac{5}{3}x^{4} - 2x^{3} + 1 \]This expression represents the slope of the tangent to the curve at any point \( x \). Solving \( f'(x) = 0 \) gives us the x-coordinates where the function's graph has horizontal tangents.
Studying the derivative provides:

• Insight into where the function increases or decreases.
• Information on potential turning points, which can be analyzed for extrema or points of inflection.

The derivative is an essential tool in calculus for comprehending the diverse behaviors of mathematical functions.

Relative Extrema

Relative extrema refer to the points on a graph where a function reaches a local maximum or minimum. To find these points, we first identify the critical points from where the derivative equals zero or is undefined.
For the polynomial \[ f(x) = -\frac{1}{3}x^{5} - \frac{1}{2}x^{4} + x \]once the critical points are found by solving \[ f'(x) = 0 \]we use these values to look for changes in the derivative's sign.
Analyzing sign changes helps in identifying:

• Relative Minimum: Occurs when the derivative changes from negative to positive, indicating a 'valley'.
• Relative Maximum: Occurs when the derivative changes from positive to negative, indicating a 'peak'.

Graphing the function is also an effective visual method to estimate relative extrema, by looking for peaks and valleys on the curve.
Relative extrema provide valuable insight into the function's local behavior and are crucial for interpreting and predicting real-world scenarios modeled by polynomials.