How comapre `1/sqrt11+1/sqrt13 ` and `2/sqrt12` in calculus?
You should use the following function to evaluate the relation between the numbers `1/sqrt11 + 1/sqrt12 and2/sqrt12` .
The function that should be a model quite well is `f(x) = 1/sqrtx - 1/sqrt(x+1)`
You need to find the derivative of this function to check if it decreases or it increases on `(0;oo).`
You should use the quotient rule to find the derivative:
`f'(x) = (-1/2sqrtx)/x+ (1/2sqrt(x+1))/(x+1)`
`f'(x) = -1/2xsqrtx + 1/2(x+1)sqrt(x+1)`
Since `2xsqrtx lt 2(x+1)sqrt(x+1) =gt 1/2xsqrtx gt 1/2(x+1)sqrt(x+1) =gt f'(x)lt 0`
`f'(x)lt 0 =gtf(11)gt f(12) =gt 1/sqrt11 - 1/sqrt 12 gt 1/sqrt12 - 1/sqrt 13`
Adding `1/sqrt13` both sides yields:
`1/sqrt11 - 1/sqrt 12 + 1/sqrt 13gt 1/sqrt12 - 1/sqrt 13 + 1/sqrt 13 =gt`
`` `1/sqrt11 - 1/sqrt 12 + 1/sqrt 13gt 1/sqrt12`
`` Adding `1/sqrt12 ` both sides yields:
`1/sqrt11 - 1/sqrt 12 + 1/sqrt 13 +1/sqrt12gt 1/sqrt12 +1/sqrt12 =gt`
`` `1/sqrt11 + 1/sqrt 13gt 2/sqrt12`
Using the derivative of a model function yields the last line that proves the given inequality.