Let f(x) and g(x) are polynomial of degree 4 such that g(α)=g′(α)=g′′(α)=0. If limx→αf(x)g(x)=0, then number of different real solutions of equation f(x)g′(x)+g(x)f′(x)=0 is equal to
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Solution
g(x)=a(x−α)3(x−β) So f(x)=b(x−α)4 h(x)=f(x)g′(x)+f′(x)g(x) =ddx(f(x).g(x)) =ddx(ab(x−α)7(x−β)) =ab(x−α)6(x−α+7x−7β) =ab(x−α)6(8x−α−7β) So h(x) has two different real solutions.