Simplifying Cubic: Find Local Extrema

[ciubba]

I need to find the local extrema of

$$\pi r^2(\frac{16}{(r+.5)^2}-1)$$

which I derived and simplified to

$$\frac{16 \pi r}{(r+.5)^3}=2 \pi r$$

which simplifies to $$\frac {16 \pi r}{2 \pi r}=(r+.5)^3$$

The radius cannot be zero, so I simplified $$8=(r+.5)^3$$

I used the binomial theorem and more algebra to obtain

$$r^3+1.5r^2+.75r-7.875$$

Now I am unsure of how to simplify the cubic. Normally I would use rational roots, but I don't know how to do that with an integer constant. I need either a method of simplifying this cubic or a place where I could have simplified the derivative better.

 

[zoki85]

8 = (r+0.5)³ , why not to take 3rd root ?

 

[ciubba]

Oh my god, I can't believe I missed that.
Thanks, I guess.

 

[Mark44]

I need to find the local extrema of

$$\pi r^2(\frac{16}{(r+.5)^2}-1)$$

which I derived and simplified to

$$\frac{16 \pi r}{(r+.5)^3}=2 \pi r$$

... which you differentiated, simplified, and then set to zero.

which simplifies to $$\frac {16 \pi r}{2 \pi r}=(r+.5)^3$$

The radius cannot be zero, so I simplified $$8=(r+.5)^3$$

I used the binomial theorem and more algebra to obtain

$$r^3+1.5r^2+.75r-7.875$$

Now I am unsure of how to simplify the cubic. Normally I would use rational roots, but I don't know how to do that with an integer constant. I need either a method of simplifying this cubic or a place where I could have simplified the derivative better.

 

[zoki85]

ciuba,ciuba you messed it up more than once

 

[Ray Vickson]

I need to find the local extrema of

$$\pi r^2(\frac{16}{(r+.5)^2}-1)$$

which I derived and simplified to

$$\frac{16 \pi r}{(r+.5)^3}=2 \pi r$$

which simplifies to $$\frac {16 \pi r}{2 \pi r}=(r+.5)^3$$

The radius cannot be zero, so I simplified $$8=(r+.5)^3$$

I used the binomial theorem and more algebra to obtain

$$r^3+1.5r^2+.75r-7.875$$

Now I am unsure of how to simplify the cubic. Normally I would use rational roots, but I don't know how to do that with an integer constant. I need either a method of simplifying this cubic or a place where I could have simplified the derivative better.

Try the full-fledged "rational root theorem' as described in http://en.wikipedia.org/wiki/Rational_root_theorem --- it works. However, you need to convert to integer coefficients throughout.