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Math Help - Maximums and Minimums of an absolute value function. [Please help I am stumped]

  1. #1
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    Maximums and Minimums of an absolute value function. [Please help I am stumped]

    Find the maximum and minimum values of f(x) on the given intervals.
    f(x)=abs(x+1)+abs(x-1)

    1>x<4 Should be greater/less than or equal to, but I dont know how to type that in.

    I know how to approach normal fxns but with the absolute values I don't know. I know to just take the derivitive, set equal to zero, and solve for x, giving you the critical points. Thanks for the help!
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  2. #2
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    Quote Originally Posted by lmao View Post
    Find the maximum and minimum values of f(x) on the given intervals.
    f(x)=abs(x+1)+abs(x-1)

    1>x<4 Should be greater/less than or equal to, but I dont know how to type that in.
    !
    I assume you mean [1,4].

    We need to find the points where f(x) is not differenciable. That is when the term inside the absolute value is zero. Thus, that happens at x=-1 and x=1. Since 1<=x<=4 we have that x=1 is the only non differenciable point. But is is an endpoint.

    Now you need to take the derivative.

    The derivative of y=|x| is y'=sgn(x) for x!=0 where sgn(x)=1 is x>0 and -1 is x<0.

    Thus, the derivative of f(x) by chain rule is,

    f'(x)=sgn(x+1)+sgn(x-1)

    Note this can be zero when,

    1)sgn(x+1) = +1 and sgn(x-1) = -1.

    2)sgn(x+1) = -1 and sgn(x+1)=1.

    If #1 then, sgn(x+1) = +1 for all x in [1,4] and sgn(x-1)=-1 never on [1,4].

    If similary #2 does not ever equal to zero.

    Thus, there are no critical points inside the open interval.
    Just check the endpoints.
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