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Question

If f(x)=⎧⎪⎨⎪⎩x2+2,x<02,x=0,x+2,x>0 then which of the following is FALSE.

A
f(x) has a maximum at x=0
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B
f(x) is strictly decreasing on the left of 0
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C
f(x) is strictly increasing on the left of 0
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D
f(x) is strictly increasing on the right of 0
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Solution

The correct option is A f(x) has a maximum at x=0
f(x)=x2+2,x<02x=0x+2x>0
Here f(0)=2
limx0f(x)=2
limx0+f(x)=2
f(x)=2x,x<0
f(x)<0 for x<0
f(x) is decreasing on the left of 0.
Hence at x=0, there is no maximum.
f"(x)=2,x<0
f"(x)>0,x<0
f(x) is increasing on the left of 0.
f(x)=1,x>0
f(x) is strictly increasing on the right of 0.

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