If Rolle's theorem holds for the function $$f(x) = 2x^3 + bx^2 + cx$$, $$x \in [-1, 1]$$ at the point $$x = \frac{1}{2}$$, then $$2b + c$$ is equal to

Rolle's theorem requires three conditions for a function $$ f(x) $$ on an interval $$[a, b]$$:

1. The function must be continuous on $$[a, b]$$.
2. The function must be differentiable on $$(a, b)$$.
3. The function values at the endpoints must be equal, i.e., $$ f(a) = f(b) $$.

Given the function $$ f(x) = 2x^3 + b x^2 + c x $$ on the interval $$[-1, 1]$$, and that Rolle's theorem holds at $$ x = \frac{1}{2} $$, we know:

• At $$ x = \frac{1}{2} $$, the derivative $$ f'(x) = 0 $$.
• The function values at the endpoints are equal: $$ f(-1) = f(1) $$.

First, compute $$ f(-1) $$ and $$ f(1) $$:

For $$ x = -1 $$:

$$ f(-1) = 2(-1)^3 + b(-1)^2 + c(-1) = 2(-1) + b(1) + c(-1) = -2 + b - c $$

For $$ x = 1 $$:

$$ f(1) = 2(1)^3 + b(1)^2 + c(1) = 2(1) + b(1) + c(1) = 2 + b + c $$

Set $$ f(-1) = f(1) $$:

$$ -2 + b - c = 2 + b + c $$

Solve for $$ c $$:

Subtract $$ b $$ from both sides:

$$ -2 - c = 2 + c $$

Add 2 to both sides:

$$ -c = 4 + c $$

Subtract $$ c $$ from both sides:

$$ -c - c = 4 \implies -2c = 4 $$

Divide both sides by -2:

$$ c = -2 $$

Now, find the derivative of $$ f(x) $$:

$$ f'(x) = \frac{d}{dx}(2x^3) + \frac{d}{dx}(b x^2) + \frac{d}{dx}(c x) = 6x^2 + 2b x + c $$

At $$ x = \frac{1}{2} $$, $$ f'\left( \frac{1}{2} \right) = 0 $$:

$$ f'\left( \frac{1}{2} \right) = 6 \left( \frac{1}{2} \right)^2 + 2b \left( \frac{1}{2} \right) + c = 0 $$

Compute $$ \left( \frac{1}{2} \right)^2 = \frac{1}{4} $$:

$$ 6 \times \frac{1}{4} + 2b \times \frac{1}{2} + c = 0 $$

Simplify:

$$ \frac{6}{4} + b + c = 0 \implies \frac{3}{2} + b + c = 0 $$

Substitute $$ c = -2 $$:

$$ \frac{3}{2} + b + (-2) = 0 $$

Simplify:

$$ \frac{3}{2} - 2 + b = 0 \implies \frac{3}{2} - \frac{4}{2} + b = 0 \implies -\frac{1}{2} + b = 0 $$

Add $$ \frac{1}{2} $$ to both sides:

$$ b = \frac{1}{2} $$

Now, compute $$ 2b + c $$:

$$ 2b + c = 2 \times \frac{1}{2} + (-2) = 1 - 2 = -1 $$

Hence, the correct answer is Option C.