Proving Maximum Volume of a Right Circular Cone: Optimization Problem Solution

[thereddevils]

Homework Statement

A right circular cone of base radius r and height h has a total surface area S and volume V . Show that 9V²=r²(S²-2pi^r2S) . (i can do this part) . Hence or otherwise , show that for a fixed surface area S , the maximum volume of the cone occurs when its semi-vertical angle , theta is given by tan theta=1/2(root 2)

The Attempt at a Solution

From the proven equation ,

9V²=r²(S²-2pi^r2S)

Differentiate this wrt to r ,

dV/dr=(2S²r-8pi Sr³)/(18V)

dV/dr=0 , S=4pi r² , substitue S with the area of cone , then
tan theta=r/h=1/(2 root 2)

This is my question , how do i prove that its a maximum ?

 

[jtyler05si]

when you took the derivative of the Volume function and set it equal to zero you are finding a critical value.
There are a couple ways to test whether it is a local min or max. The second derivative test is one of them.

note: I have to check your derivation; not that I am doubting it or anything.