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March 29, 2017

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Find any critical numbers of the function. (Enter your answers as a comma-separated list.)

g(t) = t sqrt6 − t , t < 13/3


I don't even know how to start out!

  • Calculas - ,

    I will assume you meant

    g(t) = t √(6-t) = t(6-t)^(1/2)

    g'(t) = t (1/2)(6-t)^(-1/2) (-1) + (6-t)^(1/2) using the product rule
    = 3)t-4)/(2√(6-t) )
    =0 for any max/min values of the function

    t/(2(6-t)^(1/2) = (6-t)^(1/2)
    t = 2(6-t)
    t = 12 - 2t
    3t = 12
    t = 4
    so g(4) = 4√(6-4) = 4√2

    to see if there are any points of inflection
    differentiate 3(t-4)/(2√(6-t) ) , set that result equal to zero and solve for t
    (hint: there are no points of intersection)
    You should get a second derivative of 3(t-8)/(4(6-t)^(3/2 )
    which when you set equal to zero will give an answer of
    t = 8
    BUT, t=8 is outside the domain of our function, so ... No Solution.

    y-intercepts ? , let t =0
    y = 0√6 = 0 , the origin (0,0)

    any t-intercepts?
    t(6-t)^(1/2) = 0
    t = 0 or t = 6

    look at the graph .....
    http://www.wolframalpha.com/input/?i=t%286-t%29%5E%281%2F2%29

    ignore the imaginary part of the graph (the red line)
    the point (4,4√2) is a maximum point

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