Find a number in the closed interval [1/2,3/2] such that the sum of the number and its reciprocal is a.)as small as possible

Question

Find a number in the closed interval [1/2,3/2] such that the sum of the number and its reciprocal is

a.)as small as possible
b.)as large as possible

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Answers ( No )

    0
    2021-09-10T08:47:43+00:00

    Let’s assume that number as x

    so,

    the sum of the number and its reciprocal is

    S(x)=x+\frac{1}{x}

    Firstly, we will find derivative

    S'(x)=1-\frac{1}{x^2}

    now, we can set it to 0

    and then we can solve for x

    S'(x)=1-\frac{1}{x^2}=0

    x=-1,x=1

    Since, only x=1 lies on [1/2,3/2]

    so, we will consider only x=1

    now, we can plug end values and x=1 into S

    At x=1/2:

    we can plug x=1/2

    S(\frac{1}{2})=\frac{1}{2} +\frac{1}{\frac{1}{2}}

    S(\frac{1}{2})=\frac{1}{2} +2

    S(\frac{1}{2})=\frac{5}{2}

    At x=1:

    we can plug x=1

    S(1)=1 +\frac{1}{1}

    S(1)=1 +1

    S(1)=2

    At x=3/2:

    we can plug x=3/2

    S(\frac{3}{2})=\frac{3}{2} +\frac{1}{\frac{3}{2}}

    S(\frac{3}{2})=\frac{3}{2} +\frac{2}{3}

    S(\frac{3}{2})=\frac{13}{6}

    (a)

    Smallest value:

    The smallest value among them

    S(1)=2

    So, a number is 1……….Answer

    (b)

    Largest value:

    The largest value among them

    S(\frac{1}{2})=\frac{5}{2}

    So, a number is \frac{1}{2}……..Answer

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