Thread: 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

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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