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Choice D is correct. It's given that $f\left(x\right)=\left(x-14\right)\left(x+19\right)$, which can be rewritten as $f\left(x\right)=x^2+5x-266$. Since the coefficient of the $x^2$-term is positive, the graph of $y=f\left(x\right)$ in the xy-plane opens upward and reaches its minimum value at its vertex. The x-coordinate of the vertex is the value of $x$ such that $f\left(x\right)$ reaches its minimum. For an equation in the form $f\left(x\right)=a{x}^2+bx+c$, where $a$, $b$, and $c$ are constants, the x-coordinate of the vertex is $-\frac{b}{2a}$. For the equation $f\left(x\right)=x^2+5x-266$, $a=1$, $b=5$, and $c=-266$. It follows that the x-coordinate of the vertex is $-\frac{5}{2\left(1\right)}$, or $-\frac{5}{2}$. Therefore, $f\left(x\right)$ reaches its minimum when the value of $x$ is $-\frac{5}{2}$.

Alternate approach: The value of $x$ for the vertex of a parabola is the x-value of the midpoint between the two x-intercepts of the parabola. Since it’s given that $f\left(x\right)=\left(x-14\right)\left(x+19\right)$, it follows that the two x-intercepts of the graph of $y=f\left(x\right)$ in the xy-plane occur when $x=14$ and $x=-19$, or at the points $\left(14,0\right)$ and $\left(-19,0\right)$. The midpoint between two points, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Therefore, the midpoint between $\left(14,0\right)$ and $\left(-19,0\right)$ is $\left(\frac{14+\left(-19\right)}{2},\frac{0+0}{2}\right)$, or $\left(-\frac{5}{2},0\right)$. It follows that $f\left(x\right)$ reaches its minimum when the value of $x$ is $-\frac{5}{2}$.

Choice A is incorrect. This is the y-coordinate of the y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane.

Choice B is incorrect. This is one of the x-coordinates of the x-intercepts of the graph of $y=f\left(x\right)$ in the xy-plane.

Choice C is incorrect and may result from conceptual or calculation errors.