A given quantity

Let ‘h’ be the height ‘or’ length of half cylinder, ‘r’ be the radius of half cylinder and ‘d’ be the diameter.

We know that,

Volume of half cylinder (V) =

…… (1)

Now we find the Total surface area (TSA) of the half cylinder,

TSA = Lateral surface area of the half cylinder + Area of two semi - circular ends + Area of the rectangular base

From (1)

We need total surface area to be minimum and let us take the TSA as the function of r,

For maxima and minima,

Differentiating TSA again,

At

>0(Minima)

We have got Total surface area minimum for

We know that diameter is twice of radius

Thus proved.

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