Results 1 to 4 of 4

Math Help - True or False

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    9

    True or False

    If true, show or explain why. If false provide a counterexample ( that is give a specific instance where the statement is not true).

    a.) Let functions f and g along with their second derivatives exist for all x. If f and g are both concave up for all x, then f+g is concave up for all x.

    b.) (f g)' is never equal to f' * g'

    c.) The only functions with the fourth derivative y''''= sin x must be of the form y = sin x + c where c is a constant.

    d.) The nth derivative of the function y = a^x is y to nth derivative = a^x(lna)^n
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,740
    Thanks
    645
    Hello, erimat89!

    Here's some help . . .


    b) (f g)' is never equal to f'\cdot g'
    False

    Let f(x) \:=\:3x,\;\;g(x) = \frac{C}{1-x}

    Then: . fg \;=\;\frac{3Cx}{1-x}

    And: . (fg)' \;=\;3C\left[\frac{(1-x)\!\cdot\!1 - x(\text{-}1)}{(1-x)^2}\right] \quad\Rightarrow\quad (fg)' \;=\;\frac{3C}{(1-x)^2}


    We have: . \begin{array}{ccccccc}f(x) &=& 3x & \Rightarrow & f'(x) &=& 3 \\ g(x) &=&\frac{C}{1-x} & \Rightarrow & g'(x) &=& \frac{C}{(1-x)^2} \end{array}\quad\Rightarrow\quad f'g' \:=\:\frac{3C}{(1-x)^2}




    c) The only functions with the fourth derivative y''''\:=\: \sin x must be of the form: y \:= \:\sin x + c
    False

    The function could be: . y \:=\:\sin x + 2x^3- 9




    d) The n^{th} derivative of the function y \:= \:a^x is: . \frac{d^ny}{dx^n} \:=\:a^x(\ln a)^n
    True

    \begin{array}{ccc} y &=& a^x \\ \\[-3mm]<br />
\dfrac{dy}{dx} &=& a^x(\ln a) \\ \\[-3mm]<br />
\dfrac{d^2y}{dx^2} &=& a^x(\ln a)^2 \\ \\[-3mm]<br />
\dfrac{d^3y}{dx^3} &=& a^x(\ln a)^3 \\<br />
\vdots & & \vdots \end{array}
    \begin{array}{ccc}\dfrac{d^ny}{dx^n} &=& a^x(\ln a)^n  \end{array}

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2008
    Posts
    9

    Why is part b) false?

    The example you given f'g' looks to me that it's equal to (f g)' with 3C/(1-x)^2 being the derivative of both functions whether its f'g' or (f g)'
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Sep 2008
    Posts
    9

    I think i got it

    Nevermind i think i understand thanks for your help
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 7
    Last Post: October 5th 2011, 12:45 PM
  2. Replies: 1
    Last Post: October 4th 2011, 02:19 AM
  3. how come false AND false is true?
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: September 24th 2010, 07:41 AM
  4. True or False. Prove if true show counter example if false.
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 2nd 2010, 10:54 AM
  5. true or false
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 5th 2009, 10:37 PM

Search Tags


/mathhelpforum @mathhelpforum