How comapre `1/sqrt11+1/sqrt13 ` and `2/sqrt12`  in calculus?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.