determine the monoting relative extrema and concavity of the function. f(x)= 4x^(3)+9x^(2 )+12x +3

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 = 4*x**3 + 9*x**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 = 4*x**3 + 9*x**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!

Learn how to analyze the function f(x) = 4x^3 + 9x^2 + 12x + 3 for monotonicity (increasing/decreasing behavior) and concavity (concave up/down). Step-by-step solution includes finding critical points, analyzing derivatives, and determining intervals of monotonicity and concavity. Suitable for advanced high school students studying calculus.

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