# NCERT Class 12-Mathematics: Chapter –6 Application of Derivatives Part 2

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Question 2:

Water is dripping out from a conical funnel of semi-vertical angle at the uniform rate of the surface area, through a tiny hole at the vertex of the bottom. When the slant height of cone is , find the rate of decrease of the slant height of water.

If s represents the surface area, then

Therefore,

When

Question 3:

Find the angle of intersection of the curves and .

Solving the given equations, we have and

Therefore,

i.e. points of intersection are and

Further

And

At , the slope of the tangent to the curve is parallel to axis and the tangent to the curve is parallel to axis.

Angle of intersection and that of is .

Question 4:

Prove that the function is strictly decreasing on

When

Therefore,

Thus for

Hence is strictly decreasing on .

Question 5:

Determine for which values of x, the function is increasing and for which values, it is decreasing.

Now,

Since and is continuous in and. Therefore is decreasing in and is increasing in.

Note: Here f is strictly decreasing in and is strictly increasing in .

Question 6:

Show that the function has neither maxima nor minima.

Hence is a point of inflexion i.e. neither a point of maxima nor a point of minima.

is the only critical point, and has neither maxima nor minima.

Question 7:

Using differentials, find the approximate value of