# reciprocal number and rectangle with largest area

• Nov 10th 2008, 08:48 AM
thecount
reciprocal number and rectangle with largest area
find a number such that the sum of the number and its reciprocal is as small as possible.

i have no idea what this means...

and...

find the dimensions of a rectangle of the largest area that has its base on the x axis and its other two vertices above the x axis and lying on the parabola
y=8-x^2
• Nov 10th 2008, 10:25 AM
Angel
Angel (Best Solution)
This question related to maxima and minima concept so we have to use first and second derivative test.
first we have to asume that number so,

let x is required number
according the condition x + 1/x=as small as possible

now we make one function of x and it's reciprocal

f(x)=x+1/x
take first derivative of f(x)
f '(x)=1-1/x^2
put f '(x)=0 to get maximum and minimum value
1-1/x^2=0
x=+-1

so one is maximum value and one is minimum value, to find which one is minimum we have to test now second derivative test, so

f ''(x)=2/x^3
put x=1 in f ''(x)
f ''(1)=2>0 (greater than zero mean it's minimum)
f ''(-1)=-2<0 (less than zero mean it's maximum)
so x=1 is required answer .

according this you can solve easily second part.

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