Answer :

**Given:** A sphere of fixed radius r

**To prove:** the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is

**To find:** maximum volume in terms of volume of the sphere

Let R and h be the radius and height of cone respectively

**Formula used:**

The volume of the cone is given by,

B is center of the sphere

Using Pythagoras theorem in right triangle Δ BCD,

AB is also the radius of the sphere

∴ AC = AB + BC

Differentiating both sides with respect to R:

To find the maximum volume, put

Squaring both sides:

If

Differentiating both sides again:

On putting

Therefore, the volume is maximum when

**Hence, the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is**

As the radius of sphere = r

**Therefore, the Volume of cone** **times of volume of the sphere**

**OR**

**Given:** f(x) = sin 3x – cos 3x, 0 < x < π

**To find:** intervals in which function f(x) is strictly increasing or decreasing

f(x) = sin 3x – cos 3x

Differentiating with respect to x:

{∵ sin A cos B + cos A sin B = sin (A + B)}

For interval where f(x) is increasing: f’(x) > 0

We know,

sin x > 0

0 < sin x < π and 2π < sin x < 3π

Therefore,

**Therefore, f(x) is strictly increasing in**

For interval where f(x) is decreasing: f’(x) < 0

We know,

sin x < 0

π < sin x < 2π and 3π < sin x < 4π

Therefore,

**Therefore, f(x) is strictly decreasing in**

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Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

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Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1

Find the intervalRD Sharma - Volume 1