For which real value of K does the equation 2 | vedclass

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(D) Let $f(x) = 2x^3 + 3x + K$. For $f(x)$ to have two real roots in the interval $[0, 1]$,by Rolle's Theorem,there must exist at least one point $c \

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Does not exist

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(D) Let $f(x) = 2x^3 + 3x + K$. For $f(x)$ to have two real roots in the interval $[0, 1]$,by Rolle's Theorem,there must exist at least one point $c \in (0, 1)$ such that $f'(c) = 0$. Calculating the derivative: $f'(x) = 6x^2 + 3$. Since $6x^2 + 3 > 0$ for all real $x$,$f'(x)$ is never zero. Therefore,$f(x)$ is a strictly increasing function on the interval $[0, 1]$. $A$ strictly increasing function can have at most one real root in any given interval. Thus,it is impossible for the equation $2x^3 + 3x + K = 0$ to have two real roots in the interval $[0, 1]$ for any real value of $K$.

For which real value of $K$ does the equation $2x^3 + 3x + K = 0$ have two real roots in the interval $[0, 1]$?

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