Key Concepts and Formulas

• Extreme Points: A function $f(x)$ has an extreme point (local max or min) at $x_0$ if $f'(x_0) = 0$.
• Derivatives: Basic differentiation rules for polynomials.
• Solving Equations: Finding common roots of polynomial equations.

Step-by-Step Solution

Step 1: Understand the Problem and Define the Objective

The problem states that two functions, $f(x)$ and $g(x)$, have a common extreme point. This means there exists a value $x_c$ such that $f'(x_c) = 0$ and $g'(x_c) = 0$. The goal is to find the value of $a + 2b + 7$.

Step 2: Calculate the Derivatives of $f(x)$ and $g(x)$

We need to find the first derivatives of both functions.

Given: $f(x) = \frac{x^3}{3} + 2bx + \frac{ax^2}{2}$ $g(x) = \frac{x^3}{3} + ax + bx^2$

Differentiating $f(x)$ with respect to $x$: $f'(x) = \frac{d}{dx} \left( \frac{x^3}{3} + 2bx + \frac{ax^2}{2} \right) = x^2 + ax + 2b$

Differentiating $g(x)$ with respect to $x$: $g'(x) = \frac{d}{dx} \left( \frac{x^3}{3} + ax + bx^2 \right) = x^2 + 2bx + a$

Step 3: Find the Common Extreme Point

Since $x_c$ is a common extreme point, we have $f'(x_c) = 0$ and $g'(x_c) = 0$. Therefore:

1. $x_c^2 + ax_c + 2b = 0$
2. $x_c^2 + 2bx_c + a = 0$

Subtracting equation (2) from equation (1) to eliminate the $x_c^2$ term: $(x_c^2 + ax_c + 2b) - (x_c^2 + 2bx_c + a) = 0$ $ax_c - 2bx_c + 2b - a = 0$ $x_c(a - 2b) - (a - 2b) = 0$ $(x_c - 1)(a - 2b) = 0$

Since $a \neq 2b$, we must have $x_c - 1 = 0$, which implies $x_c = 1$.

Step 4: Use the Common Point to Relate $a$ and $b$

Substitute $x_c = 1$ into either $f'(x_c) = 0$ or $g'(x_c) = 0$. Let's use $f'(x_c) = 0$:

$f'(1) = (1)^2 + a(1) + 2b = 0$ $1 + a + 2b = 0$ $a + 2b = -1$

Step 5: Calculate $a + 2b + 7$

We are asked to find the value of $a + 2b + 7$. Since we found that $a + 2b = -1$:

$a + 2b + 7 = -1 + 7 = 6$

Common Mistakes & Tips

• Forgetting the Condition $a \neq 2b$: This condition is crucial. If you ignore it, you might incorrectly conclude that $x_c$ can be any value.
• Algebra Errors: Be careful when subtracting the equations and factoring. Double-check your work.
• Not Understanding Extreme Points: Make sure you understand the relationship between extreme points and the first derivative.

Summary

We found the derivatives of the given functions, used the common extreme point condition to derive a relationship between $a$ and $b$, and then calculated the value of the requested expression. The common extreme point being 1 allowed us to find that $a + 2b = -1$, which directly leads to the final answer.

The final answer is $\boxed{6}$, which corresponds to option (A).