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Question

Which of the following functions is concave down over the set of positive real numbers:

A
f(x)=2x
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B
f(x)=x12
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C
f(x)=3x+2
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D
f(x)=x3
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Solution

The correct option is B f(x)=x12
A function f(x) is concave down if f′′(x)<0.
Now, checking options:
(A) f′′(x)=(ln2)22x
(B) f′′(x)=14x3
(C) f′′(x)=0
(D) f′′(x)=6x
Only (B) has a negative second derivative for all positive real numbers.

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