​$$\text{The equation } x^3 - 3x + k = 0 \text{ will have three distinct real roots if:}$$​​

Correct option is C

​$$\textbf{Given:}\quad x^3 - 3x + k = 0\\[10pt]\text{Let,}\quad f(x) = x^3 - 3x + k\\[10pt]\text{Now,}\\ f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 0\quad \Rightarrow x = \pm 1\\[10pt]\textbf{Now evaluate the function at critical points:}\\ f(1) = 1^3 - 3(1) + k = 1 - 3 + k = k - 2\\f(-1) = (-1)^3 - 3(-1) + k = -1 + 3 + k = k + 2\\[10pt] \text{The cubic has three distinct real roots if the local maximum and minimum lie on opposite sides of the } x\text{-axis:}\\[6pt]f(1) \cdot f(-1) < 0\\[6pt]\Rightarrow (k - 2)(k + 2) < 0\\[6pt]\Rightarrow k^2 - 4 < 0\\[6pt]\Rightarrow -2 < k < 2$$​​