NCERT Class 12-Mathematics: Chapter – 6 Application of Derivatives Part 18 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question 26:

Find the points of local maxima, local minima and the points of inflection of the function . Also find the corresponding local maximum and local minimum values.

Answer:

is the point of local maxima; local maximum

is the point of local minima; local minimum

is the point of inflection.

Question 27:

A telephone company in a town has subscribers on its list and collects fixed charges of per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of one subscriber will discontinue the service. Find what increase will bring maximum profit?

Answer:

Consider that company increases the annual subscription by ₹

So, subscribes will discontinue the service.

Total revenue of company after the increment is given by

On differentiating both sides , we get

Now,

So, is maximum when

Hence, the company should increase the subscription fee by so that it has maximum profit.

Question 28:

If the straight line touches the curve , then prove that .

Answer:

We know that, if a line touches ellipse then the required condition is

Here, given equation of the line is

And

Hence proved

Question 29:

An open box with square base is to be made of a given quantity of card board of area . Show that the maximum volume of the box is cubic units.

Answer:

Let the length of side of the square base of open box be units and its height be

Now, volume of the box (V)

Volume of the Box

On differentiating both sides we get

Now,

Again, differentiating Eq. (ii) , we get

Thus we see that volume is maximum at

Maximum volume of the box,