Answer
$\frac{10\sqrt[3] 3}{\sqrt[3] {3}+1}\approx 5.9$ $ft$
Work Step by Step
The total illumination is
$I(x)$ = $\frac{3k}{x^2}+\frac{k}{(10-x)^2}$, $0$ $\lt$ $x$ $\lt$ $10$
$I'(x)$ = $-\frac{6k}{x^3}+\frac{2k}{(10-x)^3}$
$-\frac{6k}{x^3}+\frac{2k}{(10-x)^3}$ = $0$
$6k(10-x)^3$ = $2kx^3$
$3(10-x)^3$ = $x^3$
$\sqrt[3] 3(10-x)$ = $x$
$10\sqrt[3] 3-\sqrt[3] {3}x$ = $x$
$10\sqrt[3] 3$ = $\sqrt[3] {3}x+x$
$10\sqrt[3] 3$ = $(\sqrt[3] {3}+1)x$
$x$ = $\frac{10\sqrt[3] 3}{\sqrt[3] {3}+1}$ $\approx$ $5.9$ $ft$
This gives a minimum since
$I''(x)$ $\gt$ $0$ for $0$ $\lt$ $x$ $\lt$ $10$
