Q. If f(x) = (x - 1) (x - 3)(x - 4)(x - 6) + 19 for all real value of x is

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Solution:

Note: Discarte's Rule of Signs
No equation can have more +ve real roots than it has changes of sign from +ve to -ve, and from -ve to + ve.
And similarly, number of -ve roots of f(x) can not more than the number of changes of sign in f (- x).
Now given


Here, f(x) has four changes of sign. So the equation can not have more than four positive roots. And f(-x) has no changes of sign, so by Discarte rule, the equation have no relative roots. As

Thus the equation have four positive roots.