NCERT Class 12-Mathematics: Chapter –6 Application of Derivatives Part 7

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Question 26:

Let have second derivative at such that and , then is a point of ______.

Answer:

Local minima.

Question 27:

Minimum value ofif in is ____________.

Answer:

Question 28:

The maximum value of is _____.

Answer:

Question 29:

The rate of change of volume of a sphere with respect to its surface area, when the radius is cm, is______

Answer:

6.3 EXERCISE

Short Answer (S.A)

Question 1:

A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate.

Answer:

We have, rate of decrease of the volume of spherical ball of salt at any instant is surface. Let the radius of the spherical ball of the salt be .

And surface area

Hence, the radius of ball is decreasing at a constant rate.

Question 2:

If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius.

Answer:

Given: circle where its area is increasing at a uniform rate

To prove perimeter varies inversely as the radius

Explanation: Let the radius of the circle be ‘’.

Let A be the area of the circle,

Then

As per the given criteria the area is increasing at a uniform rate, then

Now substituting the value from equation (i) in above equation, we get

Now differentiating with respect to t we get

Now let P be the perimeter of the circle, then

Now differentiating perimeter with respect to t, we get

Applying the derivatives, we get

Now substituting value from equation (ii) in the above equation we get

Cancelling the like terms we get

Converting this to proportionality, we get

Hence the perimeter of the circle with given condition varies inversely as the radius.

Hence Proved

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