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Question

Let f(x)=x2+2bx+2c2 and g(x)=x22cx+b2, b,c0. If the minimum of f(x) is greater than maximum value of g(x), then the range of bc is

A
(12,12)
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B
(0,12)
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C
(12,0)
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D
(12,)
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Solution

The correct option is A (12,12)
Given:
f(x)=x2+2bx+2c2 and g(x)=x22cx+b2, b,c0
As a>0 for f(x), so the minimum
D4a=(4b28c24) =2c2b2

As a<0 for g(x), so the maximum
D4a=(4c2+4b24) =b2+c2

As minimum of f(x) is greater than maximum of g(x), so
2c2b2>b2+c2c2>2b2b2c2<12bc(12,12)

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