# Values for k for distinct roots

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• March 11th 2011, 10:20 PM
elieh
Values for k for distinct roots
Attachment 21121

I'm on question c and I'm stuck.
Help please.
• March 12th 2011, 12:24 AM
emakarov
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.