CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
Question

For the function y=11+x2, which of the following is/are correct:

A
It has a horizontal asymptote.
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
It is symmetric about x=0
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C
It is increasing xR
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
It has maxima at x=0
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
Open in App
Solution

The correct option is D It has maxima at x=0
The function is defined for all real values of x, Consequently, it has no vertical asymptotes.
Since
limx±y(x)=limx±11+x2=0,
then the graph of the function has horizontal asymptote y=0, that is the xaxis is the horizontal asymptote.
This function is even.

y(x)=11+(x)2=11+x2=y(x).
It is obvious that the function has no roots and positive for all x.
At the point x=0, its value is
y(0)=11+02=1
The first derivative:
y(x)=(11+x2)
=1(1+x2)2(1+x2)=2x(1+x2)2.

This shows that x=0 is a stationary point. When passing through this point the derivative changes sign from plus to minus. Therefore, we have a maximum at x=0. Its value is y(0)=1.
Calculate the second derivative:

y(x)=(2x(1+x2)2)
=2(1+x2)222x2x(1+x2)(1+x2)4
=8x222x2(1+x2)3=6x22(1+x2)3.

It is equal to zero at the following points:

y(x)=06x22(1+x2)3=0
2(3x1)(3x+1)(1+x2)3=0,x1=13,x2=13.

When passing through these points the second derivative changes its sign. Therefore, both points are inflection points. The function is strictly convex downward in the intervals (,13) and (13,+) and accordingly, strictly convex upward in the interval (13,13) Since the function is even, the found inflection points have the same values of y:

y(±13)=11+(±13)2=11+13=34.
Figure presents a schematic graph of the function

Hence, we can conclude clearly from the graph that it has a horizontal asymtote and it is symmetrical about x=0 or the y axis. Also it has maxima at x=0


flag
Suggest Corrections
thumbs-up
0
BNAT
mid-banner-image