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Math Help - functions and derivatives

  1. #1
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    Arrow functions and derivatives

    Hi !

    I need some help solving a function. I've attached the problem.

    Thank you !
    Attached Thumbnails Attached Thumbnails functions and derivatives-gash.jpg  
    Last edited by CaptainBlack; April 6th 2007 at 03:51 AM.
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  2. #2
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    Quote Originally Posted by Vilkku View Post
    Hi !

    I need some help solving a function. I've attached the problem.
    Thank you !
    Hello, Vilkku,

    I'm refering to the function f(x) = 2x*e^x

    a) Since e^x > 0 for all x f(x) = 0 only if x = 0
    A produkt of 2 factors is positive (that means greater than zero) if both factors have the same sign. Since e^x > 0 for all x f(x) > 0 if x > 0.
    f(x) < 0 if x < 0

    b) Use product rule:
    f'(x) = e^x * 2 + 2x * e^x = e^x(2 + 2x)
    f is increasing if f'(x) > 0. According to the considerations of a)
    f is increasing if x > -1
    f is decreasing if -∞ < x < -1
    Thus at x = -1 is a (absolute) minimum

    c) Use product rule:
    f''(x) = (2+2x)*e^x + e^x * 2 = e^x(4+2x)
    f is convex if f''(x) > 0
    According to the considerations of a)
    f''(x) > 0 if x > -2
    f is concave if f''(x) < 0. Thus
    f''(x) < 0 if -∞ < x < -2
    Therefore the function has a point of inflection at (-2, -4/e^2)

    I've added a sketch of the graph

    EB
    Attached Thumbnails Attached Thumbnails functions and derivatives-kurvdisk_vilkku.gif  
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