To analyze the function $f(x) = x^3 + 13x^2 + 34x - 48$, let's go through some key steps:

1. Determine the Roots (Zeros) of the Function

To find the roots, we set $f(x) = 0$: $x^3 + 13x^2 + 34x - 48 = 0$ Solving a cubic equation can be challenging, so we could start by using the Rational Root Theorem, synthetic division, or a numerical approach to find possible rational roots.

2. Factorization

Once we find one root (say $x = r$), we can factor the cubic polynomial as $(x - r) \times Q(x)$, where $Q(x)$ is a quadratic equation. Solving $Q(x) = 0$ would then give the remaining roots.

3. Determine the Derivative to Find Critical Points

To find the critical points (local maxima and minima), we calculate the first derivative: $f'(x) = 3x^2 + 26x + 34$ Setting $f'(x) = 0$ gives the critical points.

4. Analyze the Behavior of the Function

• Intervals of Increase and Decrease: By testing intervals around the critical points, we can determine where $f(x)$ is increasing or decreasing.
• Concavity and Points of Inflection: Taking the second derivative $f''(x) = 6x + 26$ and setting it to zero allows us to find points of inflection, where the concavity of $f(x)$ changes.

Would you like a step-by-step solution to find the roots or further assistance with specific details?

Further Exploration Questions:

1. How can the Rational Root Theorem help in solving $f(x) = 0$ for this cubic polynomial?
2. What are the practical steps for using synthetic division on this equation?
3. How does the derivative $f'(x) = 3x^2 + 26x + 34$ help determine the local extrema?
4. Why is it useful to find the points of inflection by solving $f''(x) = 6x + 26$?
5. How can graphing $f(x)$ support our understanding of its overall behavior?

Tip: For cubic equations, identifying one root simplifies the factorization process and leads to easier solutions for the remaining terms.