Find the smallest integral value of x satisfying `(x-2)^(x^(2)-6x+8) gt 1`.

Explanation

Clearly , ` x gt 2` …(i)
`(x-2)^(x^(2)-6x+8) gt 1`
` rArr (x-2)^(x^(2)-6x+8) gt (x-2)^(0)`
When ` x - 2 gt 1`
` or x gt 3`…(ii)
We have
` x^(2) - 6x+8 gt 0 `
` rArr (x-2)(x-4) gt 0 `
` rArr x lt 2 or x gt 4 ` ....(iii)
From (ii) and (iii) ,` x gt 4`
When ` x - 2 lt 1`
` or x lt 3 ` ....(iv)
We have
` x^(2) - 6x + 8 lt 0 `
` :. (x-2)(x-4) lt 0`
` :. 2 lt x lt 4` ...(v)
From (i), (iv) and (v), we have
` 2 lt x lt 3`
Thus, `x in (2, 3) cup (4, infty)`.