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`

Explanation

`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.