Use the price-demand equation to find the values of $p$ for which demand is elastic and the values for which demand is inelastic. Assume that price and demand are both positive.
$$x=f(p)=2500-4p^2$$
The values of $p$ for which demand is elastic are
(Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

Step 1 ：Given the price-demand equation $x=2500-4p^2$, we first need to find the derivative of this function.

Step 2 ：The derivative of the function $f(p)=2500-4p^2$ is $f^{\prime }(p)=-8p$.

Step 3 ：We then substitute this derivative into the elasticity formula $E=\frac{p}{x}\cdot f^{\prime }(p)$ to get $E=-\frac{8p^2}{2500-4p^2}$.

Step 4 ：The demand is elastic if $|E|>1$ and inelastic if $|E|<1$. We solve these inequalities to find the values of $p$ for which the demand is elastic and inelastic.

Step 5 ：Solving the inequality $|E|>1$, we find that the values of $p$ for which demand is elastic are $(-\mathrm{\infty },-25)\cup (-25,-\frac{25\sqrt{3}}{3})\cup (\frac{25\sqrt{3}}{3},25)\cup (25,\mathrm{\infty })$.

Step 6 ：Solving the inequality $|E|<1$, we find that the values of $p$ for which demand is inelastic are $(-\frac{25\sqrt{3}}{3},\frac{25\sqrt{3}}{3})$.

Step 7 ：$\boxed{{\displaystyle \text{Final Answer: The values of }p\text{ for which demand is elastic are }(-\mathrm{\infty },-25)\cup (-25,-\frac{25\sqrt{3}}{3})\cup (\frac{25\sqrt{3}}{3},25)\cup (25,\mathrm{\infty }).\text{ The values of }p\text{ for which demand is inelastic are }(-\frac{25\sqrt{3}}{3},\frac{25\sqrt{3}}{3}).}}$