Which of the following is a root for the function \( f(x)=2 x^{3}-\sqrt{x-\frac{1}{2}-\frac{23}{4}} \)

To solve for roots of the function \( f(x) = 2x^3 - \sqrt{x - \frac{1}{2} - \frac{23}{4}} \), we can start by simplifying the expression under the square root. The constant term simplifies as follows: \[ -\frac{1}{2} - \frac{23}{4} = -\frac{2}{4} - \frac{23}{4} = -\frac{25}{4}. \] Thus, the expression inside the square root becomes: \[ \sqrt{x + \frac{25}{4}}. \] Setting \( f(x) = 0 \), we want to find \( x \) such that: \[ 2x^3 = \sqrt{x + \frac{25}{4}}. \] Squaring both sides will lead us to a polynomial equation that we can solve for \( x \). In case of solving for roots, graphing the function or using numerical methods can also be effective. Now, carefully check for values of \( x \) that make the squared left-hand side equal to the right-hand side. Remember, being careful with squaring equations is crucial, as it may introduce extraneous roots! Good luck with your calculations!