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Q.

Let f:ℝ→(0,∞) and g:ℝ→ℝ be twice differentiable functions such that f11 and g11are continuous functions on ℝ  . Suppose f1(2)=g(2)=0,f11(2)≠0 and g'(2)≠0. If limx→2f(x)g(x)f'(x)g'(x)=1, then

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a

f has a local minimum at x=2

b

f has a local maximum at x=2

c

f'(2)>f(2)

d

f(x)-f''(x)=0 for at least one x∈R

answer is A.

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Detailed Solution

limx→2fxgxf'xg'x=1⇒limx→2f'xgx+fxg'xf''xg'x+f'xg''x=1⇒f2g'2f''2g'2=1⇒f''2>0    since   range of f is 0,∞⇒f2>0 therefore fx has minimum at x=2  and f2-f''2=0 ⇒ fx-f''x=0  for atleast one x∈R
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