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Math Help - local max and local min values

  1. #1
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    local max and local min values

    Hi

    The goal is to find the local max and local min values I get so far and then I don't know where to go - there are 2 questions - a and b

    a) f(x) = x / (1+x)^2

    f'(x) = (1-x)/(1+x)^3 according to my calculations which means that
    x is underfined at -1 and f'(0) when x = 1 so

    x= 1 is a critical point

    f(1) = 1

    but now what - is the local maximum and minimum both 1???



    b) f(x) = x^3 - 3x^2 +5

    f'(x) = 3x^2 - 6x
    f'(x) = 3x(x-2)

    so the critical points are x = 0 and x = 2

    so f(0) = 5 and f(2) = 1 but how do you know which is local max and local min since I understand that sometimes the min is greater than the max

    Help me understand - thanks

    calculus beginner
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  2. #2
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    for critical values where f(x) is defined, the following tests are used (also called the first and second derivative tests for extrema) ...

    first derivative test ...

    f(x) has a max if f'(x) changes sign from (+) to (-) at that critical value.

    f(x) has a min if f'(x) changes sign from (-) to (+) at that critical value.


    second derivative test ...

    if f''(x) > 0 at the critical value, then f(x) is concave down and a max exists at that critical value.

    if f''(x) < 0, then f(x) is concave up and a min exists at that critical value.
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