Test the continuity and differentiability of the function defined as under at x=1 and x=2.f(x)=⎧⎪⎨⎪⎩x,x<12−x,1≤x≤2−2+3x−x2x>2
A
Continuous and differentiable at both points
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B
Dis-Continuous and not -differentiable at both points
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C
Continuous at both points but differentiable at x=2 only
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D
Continuous at both points but differentiable at x=1 only
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Solution
The correct option is C Continuous at both points but differentiable at x=2 only f(x)=⎧⎪⎨⎪⎩x,x<12−x,1≤x≤2−2+3x−x2x>2 f(1−)=1 and f(1+)=1 f is continuous at x=1 f(2−)=0 and f(2+)=0 f is continuous at x=2 f′(x)=⎧⎪⎨⎪⎩1,x<1−11≤x≤23−2xx>2 f′(1−)=1 and f′(1+)=−1 f is not differential at x=1 f′(2−)=−1 and f′(2+)=−1 f is differential at x=2