Animated Solution for Mathematics - Differentiation: The real number $k$ for which the equation, $2x^3+3x+k=0$ has two distinct real roots in $[0,1]$

Imagine you are standing on a vast, rolling landscape, and you are tasked with finding a path that crosses a specific river twice. In the world of functions, this is exactly what we are doing when we ask for a cubic equation to have two distinct real roots.

We are looking for a curve that dips down, crosses the $x$-axis, turns around, and crosses it again. Let us explore the function $f(x)=2x^3+3x+k$.

To have two distinct real roots in the interval $[0,1]$, our function $f(x)$ must behave like a roller coaster. It must cross the $x$-axis at some point $a$, then turn around at a local extremum, and cross the $x$-axis again at some point $b$.

This is where Rolle's Theorem becomes our guiding light. Rolle's Theorem tells us that if a continuous and differentiable function has two roots, its derivative must be zero at some point between them.

This is the "turning point" we are looking for. If we can prove that our function never turns, we prove that it can never have two roots.

Calculus acts as our microscope here. To see if our function ever turns, we calculate its derivative, $f^{\prime }(x)$.

The derivative tells us the slope of the tangent line at any point. If the slope is positive, the function is climbing; if it is negative, the function is falling. Let us differentiate $f(x)=2x^3+3x+k$ term by term:

Using the power rule, the derivative of $2x^3$ is $6x^2$, and the derivative of $3x$ is $3$. Since $k$ is a constant, its derivative is $0$. Thus, we arrive at:

Now, look closely at this expression. We have $6x^2+3$. For any real number $x$, the square $x^2$ is always non-negative ($x^2\ge 0$).

When we multiply this by $6$, it remains non-negative. Finally, when we add $3$, the entire expression $6x^2+3$ is guaranteed to be at least $3$.

This means $f^{\prime }(x)>0$ for all real values of $x$. There is no value of $x$ where the slope is zero or negative. The function is strictly increasing.

Because our function $f(x)$ is strictly increasing, it can cross the $x$-axis at most once. It is physically and mathematically impossible for it to turn around and cross the axis a second time.

Therefore, no matter what real value we choose for $k$, the equation $2x^3+3x+k=0$ can never have two distinct real roots.

The answer is elegant and absolute: such a value of $k$ does not exist. We have successfully navigated the landscape of this cubic function and found that the path we were looking for simply cannot be built.