Application Of Derivatives

Given, and
Since, the function
is monotonically increase for all
.

Given,
On differentiating w.r.t. x, we get


Since, the tangent to the curve is perpendicular to the line




From E (i),
Hence, required point is .

Given curves is ..(i)
If a line is a tangent of given curve, then it will touch at only one point.
For to be a tangent,

On putting the value of x in given curve (i), we get







or
or
or [not integral value] So,
Then, will be the tangent to the given curve.

Given,


..(i)
For maxima or minima,


Again, differentiating E (i) . r. t, x. we get


At
At and at
So, at
has neither maximum not minimum value At has maximum value
At has minimum value

The correct answer is D:(3,2)
Given curve is
Differentiating w.r.t. , we get


Let Slope of normal at
Also, slope of line is
Since the line and normal to the curve are parallel.




Since lies on




or
So, point is or .

We have,
Differentiating (i) w.r.t. x, we get


Now,
and
Sign change for

Hence, is decreasing in