# NCERT Class 12-Mathematics: Chapter – 6 Application of Derivatives Part 20 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 32**:

AB is a diameter of a circle and is any point on the circle. Show that the area of is maximum, when it is isosceles.

**Answer**:

We have,

And

Let

Now, area of

Now, differentiating both sides we get

Now,

(change to in figure)

Again, differentiating both sides , we get

At

For the area of triangle is maximum.

For

Since,

Hence, the triangle is isosceles.

**Question 33**:

A metal box with a square base and vertical sides is to contain . The material for the top and bottom costs and the material for the side՚s costs . Find the least cost of the box.

**Answer**:

**Question 34**:

The sum of the surface areas of a rectangular parallelepiped with sides and and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.

**Answer**: