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Math Help - Values for k for distinct roots

  1. #1
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    Values for k for distinct roots

    Values for k for distinct roots-183423_10150110998344792_587609791_5933819_7730839_n.jpg

    I'm on question c and I'm stuck.
    Help please.
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  2. #2
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    The equation k = |f(x)| may have 2, 4, 6, 7 and 8 roots. Some of these numbers of roots are obtained when k is not an integer, though.

    Let -m be the minimum of (x^2 - 4)(x^2 - 25), i.e., -m/\lambda is the minimum of f(x). According to my calculations, m > f(0) (0 is one of the critical points of f(x)). Therefore, when k > m/\lambda, the equation has two roots. From the inequality 2 > m/\lambda you can find the range of \lambda for k = 2. Then find whether 4 = m/\lambda gives an integer \lambda. When f(0) = 100/\lambda < k < m/\lambda, the equation has 6 roots. Solving 100/\lambda < 6 < m/\lambda gives you the values of \lambda for k = 6, and so on.
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