Solve xi^prime(r)=1/3pih(r_1^2+r_2^2+r_{1}r_{2})

$$\xi ^ { \prime } ( r ) = \frac { 1 } { 3 } \pi h ( r _ { 1 } ^ { 2 } + r _ { 2 } ^ { 2 } + r _ { 1 } r _ { 2 } )$$

Solve for h

$$\left\{\begin{matrix}\\h=0\text{, }&\text{unconditionally}\\h\in \mathrm{R}\text{, }&r_{1}=0\text{ and }r_{2}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{3}\pi h by r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\xi )r=\frac{1}{3}\pi hr_{1}^{2}+\frac{1}{3}\pi hr_{2}^{2}+\frac{1}{3}\pi hr_{1}r_{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{3}\pi hr_{1}^{2}+\frac{1}{3}\pi hr_{2}^{2}+\frac{1}{3}\pi hr_{1}r_{2}=\frac{\mathrm{d}}{\mathrm{d}x}(\xi )r$$

Combine all terms containing h.

$$\left(\frac{1}{3}\pi r_{1}^{2}+\frac{1}{3}\pi r_{2}^{2}+\frac{1}{3}\pi r_{1}r_{2}\right)h=\frac{\mathrm{d}}{\mathrm{d}x}(\xi )r$$

The equation is in standard form.

$$\frac{\pi r_{1}^{2}+\pi r_{1}r_{2}+\pi r_{2}^{2}}{3}h=0$$

Divide 0 by \frac{1}{3}\pi r_{1}^{2}+\frac{1}{3}\pi r_{2}^{2}+\frac{1}{3}\pi r_{1}r_{2}.

$$h=0$$

Solve for r

$$r\in \mathrm{R}$$

$$\left(r_{1}=0\text{ and }r_{2}=0\right)\text{ or }h=0$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{3}\pi h by r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\xi )r=\frac{1}{3}\pi hr_{1}^{2}+\frac{1}{3}\pi hr_{2}^{2}+\frac{1}{3}\pi hr_{1}r_{2}$$

The equation is in standard form.

$$0=\frac{\pi hr_{1}^{2}+\pi hr_{1}r_{2}+\pi hr_{2}^{2}}{3}$$

This is false for any r.

$$r\in$$