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Question

Let f be any function continuous on [a,b] and twice differentaible on (a,b). If for all x(a,b), f(x)>0 and f′′(x)<0, then for any c(a,b),f(c)f(a)f(b)f(c) is greater than :

A
bcca
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B
1
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C
cabc
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D
b+aba
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Solution

The correct option is C cabc
f is continuous on [a,b] and twice differentaible on (a,b).

LMVT is applicable
For p(a,c), f(p)=f(c)f(a)ca
For q(c,b), f(q)=f(b)f(c)bc

f′′(x)<0f(x) is decreasing

f(p)>f(q)
f(c)f(a)ca>f(b)f(c)bc

f(c)f(a)f(b)f(c)>cabc
(as f(x)>0f(x) is increasing)

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