MTH101 Assignment 1 Solution and Discussion


  • Cyberian's Gold

    ASSIGNMENT#1
    Fall 2019

    Total marks: 10
    Due Date: 30th December, 2019

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    Question:
    If 5a805f3a-95f4-410e-8c81-dd2e9c2d70d1-image.png Find the intervals where the given function is

    1. Increasing
    2. Decreasing
    3. Concave up
    4. Concave down respectively.
      Also find the point of inflection for the given function.

  • Cyberian's Gold


  • Cyberian's Gold

    Q: If Find the intervals where the given function is

    Solution:

    Increasing
    

    f(x)=x3-27x

    After Taking Derivative

    f `(x)=3x2-27

    Function will increase if it is greater than zero

    f `(x)=3x2-27>0

    3x2-27>0

    x2-9>0

    x2>9

    After taking square root

    x>3

    So, function increase on (3,∞)

    Decreasing
    

    f(x)=x3-27x

    Take Derivative

    f `(x)=3x2-27

    Function will decrease when it is less than zero

    f `(x)=3x2-27<0

    3x2-27<0

    x2-9<0

    x2<9

    After taking square root

    x<3

    So, function decrease on (- ∞, 3)

    Concave up
    

    f(x)=x3-27x

    After Taking Derivative

    f `(x)=3x2-27

    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)=x3-27x

    After Taking Derivative

    f `(x)=3x2-27

    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)=x3-27x ------------------ (1)

    After Taking Derivative

    f `(x)=3x2-27

    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)



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