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calculus

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A) How do you prove that if 0(<or=)x(<or=)10, then 0(<or=)sqrt(x+1)(<or=)10?

B) So once that is found, then how can you prove that if 0(<or=)u(<or=)v(<or=)10, then 0(<or=)sqrt(u+1)(<or=)sqrt(v+1)(<or=)10?

  • calculus - ,

    How do you prove that if 0(<or=)x(<or=)10, then 0(<or=)sqrt(x+1)(<or=)10?
    ===========================
    does the square root increase (is the derivative positive) as x goes from 0 to 10 ?
    If so the left side of the domain is minimum and the right side is maximum of the function and we only need to test the ends.

    d (x+1)^.5 / dx = .5 /sqrt(x+1)
    that is positive everywhere in the domain so all we have to prove is the end points.

    0 </= x </= 10

    if x = 0
    sqrt x+1 = sqrt 1 = 1
    if x = 10
    sqrt x+1 = sqrt 11 = 3.32

    so
    1 </ sqrt(x+1) </= 3.32

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