To analyze the function $f(x) = 4x^3 + 9x^2 + 12x + 3$ for monotonicity (increasing/decreasing behavior), relative extrema (local maxima and minima), and concavity (concave up/down), we'll follow these steps:

1. Find the First Derivative $f'(x)$

The first derivative will help us determine the critical points, where the function's slope is zero, and analyze the monotonicity.

2. Find the Critical Points

Set $f'(x) = 0$ to find the critical points, which may correspond to relative maxima, minima, or points of inflection.

3. Analyze Monotonicity

Use the sign of $f'(x)$ to determine where the function is increasing or decreasing.

4. Find the Second Derivative $f''(x)$

The second derivative will help us determine the concavity and find points of inflection.

5. Analyze Concavity

Use the sign of $f''(x)$ to determine where the function is concave up or down.

Let's perform these calculations step by step.import sympy as sp

Define the variable and function

x = sp.symbols('x') f = 4x**3 + 9x**2 + 12*x + 3

Step 1: Find the first derivative

f_prime = sp.diff(f, x)

Step 2: Find the critical points by setting f'(x) = 0

critical_points = sp.solve(f_prime, x)

Step 3: Find the second derivative

f_double_prime = sp.diff(f_prime, x)

Step 4: Determine the nature of critical points (concave up or down)

concavity_at_critical_points = [(point, f_double_prime.subs(x, point)) for point in critical_points]

Step 5: Analyze the intervals for monotonicity

increasing_intervals = sp.solve(f_prime > 0, x) decreasing_intervals = sp.solve(f_prime < 0, x)

Output the results

f_prime, critical_points, f_double_prime, concavity_at_critical_points, increasing_intervals, decreasing_intervalsimport sympy as sp

Define the variable and function

x = sp.symbols('x') f = 4x**3 + 9x**2 + 12*x + 3

Step 1: Find the first derivative

f_prime = sp.diff(f, x)

Step 2: Find the critical points by setting f'(x) = 0

critical_points = sp.solve(f_prime, x)

Step 3: Find the second derivative

f_double_prime = sp.diff(f_prime, x)

Step 4: Determine the nature of critical points (concave up or down)

concavity_at_critical_points = [(point, f_double_prime.subs(x, point)) for point in critical_points]

Step 5: Analyze the intervals for monotonicity

increasing_intervals = sp.solve(f_prime > 0, x) decreasing_intervals = sp.solve(f_prime < 0, x)

Output the results

f_prime, critical_points, f_double_prime, concavity_at_critical_points, increasing_intervals, decreasing_intervalsIt seems like I can’t do more advanced data analysis right now. Please try again later. If you need help with the calculations, you can also try doing them manually or using a different tool. If you have any other questions, feel free to ask!