Q. An open topped box is to be constructed by removing equal squares from each corner of a by rectangular sheet of aluminium and folding up the sides. The largest volume of such a box is

Application of Derivatives Report Error

A

B

C

D

Solution:

Let be the length of a side of the removed squares. Then, the height of the box is , length is and breadth is . If is the volume of the box, then

Therefore,

Now, gives . But
Thus, we have . Now .
Therefore, is the point of maxima, i.e., if we remove a square of side from each corner of the sheet and make a box from the remaining sheet, then the volume of the box such obtained will be the largest and it is given by

Questions from Application of Derivatives

1. Let
• Column 1 contains information about zeros of and .
• Column 2 contains information about the limiting behavior of and at infinity.
• Column 3 contains information about increasing/decreasing nature of and .

Column 1 Column 2 Column 3

(I) for some (i) (P) is increasing in ()

(II) for some (ii) (Q) is decreasing in ()

(III) for some (ii) (R) is increasing in ()

(IV) for some (ii) (S) is decreasing in ()

Which of the following options is the only INCORRECT combination?

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2. The angle of intersection between the curves and where denotes the greatest integer is

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3. Let be a positive increasing function with
Then

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4. The slope of the normal to the curve at the point is

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5. The normal to the curve at the point where the curve intersects the y-axis passes through the point :

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