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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`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.