MTH101 Assignment 1 Solution and Discussion

ASSIGNMENT#1
Fall 2019Total marks: 10
Due Date: 30th December, 2019DON’T MISS THESE: Important instructions before attempting the solution of this assignment:
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If Find the intervals where the given function is Increasing
 Decreasing
 Concave up
 Concave down respectively.
Also find the point of inflection for the given function.


Q: If Find the intervals where the given function is
Solution:
Increasing
f(x)=x327x
After Taking Derivative
f `(x)=3x227
Function will increase if it is greater than zero
f `(x)=3x227>0
3x227>0
x29>0
x2>9
After taking square root
x>3
So, function increase on (3,∞)
Decreasing
f(x)=x327x
Take Derivative
f `(x)=3x227
Function will decrease when it is less than zero
f `(x)=3x227<0
3x227<0
x29<0
x2<9
After taking square root
x<3
So, function decrease on ( ∞, 3)
Concave up
f(x)=x327x
After Taking Derivative
f `(x)=3x227
After taking second derivative
f ``(x)=6x
f ``(x)=6x>0
6x>0
x>0
So, it is concave up on (0,+∞)
Concave down
f(x)=x327x
After Taking Derivative
f `(x)=3x227
After taking second derivative
f ``(x)=6x
f ``(x)=6x<0
6x<0
x<0
So, it is concave down on ( ∞,0)
POINT OF INFLECTION
f(x)=x327x  (1)
After Taking Derivative
f `(x)=3x227
After taking second derivative
f ``(x)=6x
f ``(x)= 6x =0
6x=0
x=0
Put this in (1)
f(0)=(0)3 27(0)
f(0)=0
So, inflection point is (0 , 0)