Key Concepts and Formulas

• Derivative and Monotonicity: If $f'(x) > 0$ on an interval, then $f(x)$ is strictly increasing on that interval. If $f'(x) < 0$, then $f(x)$ is strictly decreasing.
• Number of Roots: A strictly monotonic function can have at most one real root. For a continuous function to have two distinct real roots in an interval, it must have at least one local extremum within that interval.
• Intermediate Value Theorem (IVT): If $f(x)$ is continuous on $[a, b]$ and $N$ is a number between $f(a)$ and $f(b)$, then there exists a $c$ in $(a, b)$ such that $f(c) = N$.

Step-by-Step Solution

Step 1: Define the function and find its derivative. We are given the equation $2x^3 + 3x + k = 0$. Let's define the function $f(x) = 2x^3 + 3x + k$. We want to find the values of $k$ for which $f(x) = 0$ has two distinct real roots in the interval $[0, 1]$. First, we find the derivative of $f(x)$ with respect to $x$: $f'(x) = \frac{d}{dx}(2x^3 + 3x + k) = 6x^2 + 3$

Step 2: Analyze the monotonicity of the function. Now, we analyze the sign of $f'(x)$. Since $x^2 \ge 0$ for all real numbers $x$, we have $6x^2 \ge 0$. Therefore, $6x^2 + 3 \ge 3 > 0$. This means $f'(x) > 0$ for all $x \in \mathbb{R}$. Since $f'(x) > 0$ for all $x$, the function $f(x)$ is strictly increasing on the entire real line, and in particular, on the interval $[0, 1]$.

Step 3: Determine the possible number of roots. Because $f(x)$ is strictly increasing, it can have at most one real root. Therefore, it is impossible for $f(x)$ to have two distinct real roots in the interval $[0, 1]$.

Step 4: Conclusion Since the function is strictly increasing, it can have at most one root. Therefore, there is no value of $k$ for which the given equation has two distinct real roots in the interval $[0, 1]$.

Common Mistakes & Tips

• Misinterpreting Monotonicity: A common mistake is to assume that a function with a derivative that is always positive must have a root. A strictly increasing function can lie entirely above or entirely below the x-axis.
• Forgetting the Definition of Distinct Roots: Remember that distinct roots must be different from each other.
• Checking for Local Extrema: When looking for multiple roots, always check for local maxima or minima within the interval of interest.

Summary

We analyzed the function $f(x) = 2x^3 + 3x + k$ by finding its derivative and determining its monotonicity. Since the derivative is always positive, the function is strictly increasing, meaning it can have at most one real root. Therefore, there is no value of $k$ for which the equation $2x^3 + 3x + k = 0$ has two distinct real roots in the interval $[0, 1]$.

Final Answer The final answer is \boxed{does not exist}, which corresponds to option (D).