InterviewSolution
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InterviewSolution
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Find the largest natural number a for which the maximum value of `f(x)=a-1+2x-x^2`is smaller thante ninimum value of `g(x)=x^2-2a x=10-2adot` |
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Answer» `f(x) = a - 1 + 2x - x^(2)` `= a - (x^(2)-2x + 1)` `= a (x - 1) ^(2)` Hence, the maximum value of `f(x)` is a when `(x - 1)^(2) = 0 or x = 1` `g(x) = x^(2) - 2ax + 10 -2a` `=(x - a)^(2) + 10 - 2a - a^(2)` Hence, the minimum value of `g(x) is 10 -2a - a^(2)` when `(x - a)^(2) = 0 or x= a`. Now given that maximum of `f(x)` is smaller than the minimum of g(x) . Thus, `a lt - a^(2)+ 10 - 2a` or `a^(2) + 3a - 10 lt 0` or ` (a + 5) (a - 2) lt 0` `therefore -5 lt a lt 2` So, The largest natural number a = 1. |
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