InterviewSolution
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InterviewSolution
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y= f(x) is a polynomial function passing through point (0, 1) and which increases in the intervals (1, 2) and (3, oo) and decreases in the intervals (oo,1) and (2, 3). If f(x)=0 has four real roots, then the range of values of leading coefficient of polynomial is |
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Answer» [4/9, 1/2] HENCE, `f'(x)=a(x-1)(x-2)(x-3), a gt 0` `implies f(x)=INT a(x^(3)-6x^(2)+11x-6)dx` `=a((x^(4))/(4)-2x^(3)+(11x^(2))/(2)-6x)+C` Also, `f(0)=1 implies c=1` ` :. f(x)=a((x^(4))/(4)-2x^(3)+(11x^(2))/(2)-6x)+1 "(1)" ` So, graph is symmetrical about line `x=2` and range is `[f(1), oo) or [f(3),oo).` For `f(x)=0, " we have " (x^(4))/(4)-2x^(3)+(11x^(2))/(2)-6x= -(1)/(a)` For `x=1, -(9)/(4) = -(1)/(a) " or " a=(4)/(9)` For `x=2, -2= -(1)/(a) " or " a=(1)/(2)` So, `f(x)=0` has FOUR roots if `a in [(4)/(9),(1)/(2)]`. |
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