Given,

• The volume of the cone.

• The cone is right circular cone.

• The cone has least curved surface.

Let us consider,

• The radius of the circular base be ‘r’ cms.

• The height of the cone be ‘h’ cms.

• The slope of the cone be ‘l’ cms.

Given the Volume of the cone = πr²l

----- (1)

The Surface area cylinder is = πrl

S = πrl

[substituting (1) in the Surface area formula]

[squaring on both sides]

----- (2)

For finding the maximum/ minimum of given function, we can find it by differentiating it with r and then equating it to zero. This is because if the function Z has a maximum/minimum at a point c then Z’(c) = 0.

Differentiating the equation (2) with respect to r:

[Since and ]

------- (3)

To find the critical point, we need to equate equation (3) to zero.

---- (4)

Now to check if this critical point will determine the minimum surface area of the cone, we need to check with second differential which needs to be positive.

Consider differentiating the equation (3) with r:

[Since and ]

Now let us find the value of

As , so the function Z = S² is minimum

Now consider, the equation (4),

Now substitute the volume of the cone formula in the above equation.

π²r⁴h² = 2 π²r⁶

2r² = h²

Hence, the relation between h and r of the cone is proved when S is the minimum.