The function given is:

$f(x) = x^3 + 4x^2 - 5x + 2$

Key Steps for Analyzing this Function

1. Finding Roots: Solve $f(x) = 0$ to find the x-intercepts or real roots.
2. First Derivative: To analyze the behavior of $f(x)$ (increasing or decreasing intervals and critical points), compute $f'(x)$: $f'(x) = 3x^2 + 8x - 5$
3. Second Derivative: To find concavity and inflection points, compute $f''(x)$: $f''(x) = 6x + 8$
4. Finding Intervals of Concavity: Set $f''(x) = 0$ and solve for $x$.
5. End Behavior: Since this is a cubic function, as $x \to \infty$, $f(x) \to \infty$ and as $x \to -\infty$, $f(x) \to -\infty$.

Would you like to go into the details of these steps, or do you have specific questions?

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Related Questions

1. What are the real roots of $f(x) = x^3 + 4x^2 - 5x + 2$?
2. How do we find the critical points and classify them?
3. What are the intervals where $f(x)$ is concave up or concave down?
4. How does the end behavior of $f(x)$ relate to the degree and leading coefficient of the polynomial?
5. How would we approximate the x-intercepts using numerical methods?

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Tip: Remember, cubic functions have at least one real root and often change concavity at one point, which can be an inflection point.