5. Find the minimum distance from the point $(0,5 / 2)$ to the graph of $y=x^{4} / 8$.
Solution: $\sqrt{17} / 2$
We want to minimize $x^{2}+\left(x^{4} / 8-5 / 2\right)^{2}=x^{8} / 64-5 x^{4} / 8+x^{2}+25 / 4$, which is equivalent to minimizing $z^{4} / 4-10 z^{2}+16 z$, where we have set $z=x^{2}$. The derivative of this expression is $z^{3}-20 z+16$, which is seen on inspection to have 4 as a root, leading to the factorization $(z-4)(z+2-2 \sqrt{2})(z+2-2 \sqrt{2})$. Since $z=x^{2}$ ranges over $[0, \infty)$, the possible minima are at $z=0, z=-2+2 \sqrt{2}$, and $z=4$. However, the derivative is positive on $(0,-2+2 \sqrt{2})$, so this leaves only 0 and 4 to be tried. We find that the minimum is in fact achieved at $z=4$, so the closest point on the graph is given by $x= \pm 2$, with distance $\sqrt{2^{2}+\left(2^{4} / 8-5 / 2\right)^{2}}=\sqrt{17} / 2$.