Question

# Let f(x)=⎧⎪⎨⎪⎩5x+1,              x≤2x∫0(5+|1−t|) dt,  x>2 then which of the following statement(s) is /are incorrect?f(x) is continuous but not differentiable at x=2f(x) is not continuous at x=2f(x) is differentiable for all x∈Rf(x) is differentiable for x>2

Solution

## The correct options are B f(x) is not continuous at x=2 C f(x) is differentiable for all x∈RFor x>2, f(x)=x∫0(5+|1−t|) dt        =1∫0(5+|1−t|) dt+x∫1(5+|1−t|) dt        =1∫0(6−t) dt+x∫1(4+t) dt        =[6t−t22]10+[4t+t22]x1        =6−12+4x+x22−4−12        =x22+4x+1 ∴f(x)=⎧⎨⎩5x+1,             x≤2x22+4x+1,   x>2 At x=2,  f(2)=11 limx→2−f(x)=11 limx→2+f(x)=2+8+1=11 So f(x) is continuous at x=2. f′(x)={5,              x≤2x+4,       x>2 At x=2 limx→2−f′(x)=5 limx→2+f′(x)=2+4=6 limx→2−f′(x)≠limx→2+f′(x) So f(x) is not differentiable at x=2 Therefore f(x) is differentiable for all x∈R−{2}

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