Let [x] denotes the greatest integer function and f (x) = max {1 + x + [x], 2 + x, x + 2[x]}, $0 \leq x \leq 2$ let m be the number of Points in [0, 2]where f is not continuous and n be the number of points in (0, 2) Where f is not differentiable, then (m + n)² + 2 is equal to

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Detailed Solution

$\begin{array}{l} f (x) = \{\begin{cases} max {x + 1, x + 2, x} & 0 \leq x < 1 \\ max {x + 2, x + 2, x + 2} & 1 \leq x < 2 \\ max {5, 4, 6} & x = 2 \end{cases} \\ \Rightarrow f (x) = \{\begin{cases} x + 2 & 0 \leq x < 1 \\ x + 2 & 1 \leq x < 2 \\ 6 & x = 2 \end{cases} \end{array}$
max $f (x) = \{\begin{cases} 2 + x if 0 \leq x < 2 \\ 6 if x = 2 \end{cases}$
Discontinuous at x=2 then m=1
Differentiable at every in (0, 2) then n = 0 (m + n)² + 2 = 3