Application Of Derivatives

Since,

Equation of the tangent at is

Put and , we get

It is clear from the graph that both the curves have a tangent at the coordinate axes, so the angle between the curves is

Given function is,


Now, sign scheme of is

So, is increasing in and and decreasing in .

Given curve is, ...(i)
On differentiating w.r.t , we get



Since, tangent are parallel to -axis only.
i.e.,

...(ii)
Now, on putting the valus of from Eqs. (i) in (ii), we get



From E (ii) When
and when
Hence required points are and

Given, and
We know that

The point of intersection of given curve is


Point of intersection is
Now,


At point


Slope of line is
Angle between two curves is

Given equation of parabola is

The point at which the tangent to the curve is parallel to the line is the nearest point on the curve.
On differentiating both sides of E (i), we get


Also, slope of the given line is .


From E (i),

Hence, the required point is

We have,



has no real root.
has not more than one real root.