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Question

Let f:[0,1]R (the set of all real numbers) be a function. Suppose the function f is twice differentiable,f(0)=f(1)=0 and satisfies f′′(x)2f(x)+f(x)ex, x[0,1]
If the function ex f(x) assumes its minimum in the interval [0, 1] at =14, which of the following is true?

A
f(x)<f(x),14<x<34
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B
f(x)<f(x),14<x<14
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C
f(x)<f(x),0<x<14
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D
f(x)<f(x),34<x<1
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Solution

The correct option is C f(x)<f(x),0<x<14
g(x)=exf(x)
g(x)=exf(x)exf(x)=exf(x)f(x)
As x=14 is point of local minima in [0, 1]
g(x)<0 for xϵ(0,14)
and g(x)>0 for xϵ(14,1)
In(0,14),g(x)<0
ex(f(x)f(x))<0f(x)<f(x)

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