Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = (x⁴/2) - 3x² + 4x + 1

Verified step by step guidance

1. Identify the function: \( f(x) = \frac{x^4}{2} - 3x^2 + 4x + 1 \). This is a polynomial function of degree 4.
2. Find the derivative \( f'(x) \) to determine critical points and analyze the behavior of the function. Use the power rule to differentiate: \( f'(x) = 2x^3 - 6x + 4 \).
3. Set \( f'(x) = 0 \) to find critical points: \( 2x^3 - 6x + 4 = 0 \). Solve this equation to find the values of \( x \) where the function's slope is zero, indicating potential maxima, minima, or points of inflection.
4. Find the second derivative \( f''(x) \) to analyze concavity: \( f''(x) = 6x^2 - 6 \). Use \( f''(x) \) to determine intervals where the function is concave up or concave down, and identify any points of inflection.
5. Evaluate the function \( f(x) \) at the critical points and endpoints of the interval to determine the local and global extrema. Use these values to sketch the graph, ensuring to check your work with a graphing utility.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function given, ƒ(x) = (x⁴/2) - 3x² + 4x + 1, is a polynomial of degree four, which means its highest exponent is four. Understanding the general shape and behavior of polynomial functions is crucial for graphing them accurately.

Graphing Techniques

Graphing techniques involve methods to visualize functions on a coordinate plane. This includes identifying key features such as intercepts, turning points, and asymptotic behavior. For polynomial functions, one can find critical points by taking the derivative, which helps in determining where the function increases or decreases, aiding in sketching the graph.

Using Graphing Utilities

Graphing utilities are software or online tools that allow users to plot functions quickly and accurately. They can provide visual representations of functions, making it easier to verify the results obtained through manual graphing. Utilizing these tools can enhance understanding of the function's behavior and confirm the accuracy of the graph created by hand.