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Homework Help: Max & Min

  1. Mar 10, 2010 #1
    Show that the function f(x) = x^21 + x^11 + 13x does not have a local maximum or minimum.

    So f '(x) = 21x^20 + 11x^10 + 13.

    My reasoning is as follows:

    Since the exponents (10 and 20) are even, 21x^20 and 11x^10 can never be negative, and thus, summing them can never produce a negative number to make the expression 0 = 21x^20 + 11x^10 + 13 true. So there are no critical numbers, and therefore no local max or min.

    Would this be correct?
     
  2. jcsd
  3. Mar 10, 2010 #2
    Yes, since for stationary/critical/etc... points to exist, your function's derivative has to have points in which its value is 0. Since your function can never have 0 values, you're correct.
    The graphical interpretation is also quite neat. Try these in Mathematica, it'll all be clear in a second, and you can also use it in the case of more complicated functions:

    [tex]Plot[x^{21} + x^{11} + {13*x}, \{ x, -10, 10\\\}] [/tex]

    [tex]Plot[21*x^{20} + 11*x^{10} + 13*x, \{ x, -10, 10\\\}][/tex]
     
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