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Solution


Q.

Let [x] denote the greatest integer function and f(x)=max{1+x+[x], 2+x, x+2[x]}, 0≤x≤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        [2023]
1 2  
2 3  
3 6  
4 11  

Ans.

(2)

In [0,1]

f(x)=max{1+x,2+x,x}=2+x

In (1, 2)

f(x)=max{1+x+1,2+x,x+2}=2+x

At x = 2

f(x)=max{1+x+2,2+x,x+(2×2)}   

=max{x+3,x+2,x+4}   

=x+4

In [0, 2], f(x) is not continuous at x = 2

In (0, 2), f(x) is not differentiable function.

∴  m=1,  n=0

So, (m+n)2+2=(1+0)2+2=1+2=3